blogs.sch.grblogs.sch.gr/patsimas/files/2014/01/oria_4.pdf · 163 oo D E x 0 x 0 1 x 1 x 1 limf x...
136
0 x 3.43. x 3 lim f x 2 x 3 lim f x 1 , x 3 lim f x . x 5 limf x 4 f5 2 4 . x 6 x 6 lim f x lim f x 5 . x 2 x 2 limf x lim f x 1 x 2 lim f x f 0,2 . x 0 x 0 limf x lim f x 2 . x 12 lim f x f D f D 3.44. : 2 x x 6 0 x 3 x 2 0 x 3 x 2 f D 3,2 . 2 3 2 2 x 1 x x 6 x 7x 6 fx x 1 x x 6 x x 6 . x 2 limf x 1 x 3 lim f x 4 . 3.45. 2 x 6x 0 x ,0 6, 2 2 x 16 0 x 16 x 4 x ,4 4, . f A ,4 6, . f x 3 limf x 3.46. 2 2 2 5 8 0 2 5 6 2 1 4 5 4 0 3.47. 4 4 x 1 x 1 3 3 x 1 lim[f x gx 1] f x gx 1 lim fx 7 3g x lim[ fx 7 3g x ] 4 x 1 x 1 3 x 1 x 1 limf x lim g x 1 lim fx 7 lim 3g x
Transcript of blogs.sch.grblogs.sch.gr/patsimas/files/2014/01/oria_4.pdf · 163 oo D E x 0 x 0 1 x 1 x 1 limf x...
0x
3.43.
x 3
lim f x 2
x 3
lim f x 1, x 3lim f x .
x 5lim f x 4 f 5 2 4 .
x 6 x 6
lim f x lim f x 5 .
x 2 x 2
limf x lim f x 1 x 2
lim f x f
0,2 .
x 0 x 0
limf x lim f x 2 .
x 12lim f x
fD
fD
3.44. : 2x x 6 0 x 3 x 2 0 x 3
x 2 f
D 3,2 .
23
2 2
x 1 x x 6x 7x 6f x x 1
x x 6 x x 6.
x 2lim f x 1
x 3lim f x 4 .
3.45. 2
x 6x 0 x ,0 6,
2 2x 16 0 x 16 x 4 x , 4 4, .
fA , 4 6, . f
x 3
lim f x
3.46.
2
2
2
5 8 02 5 6 2 1 4
5 4 0
3.47.
44
x 1
x 1 3 3
x 1
lim[f x g x 1]f x g x 1lim
f x 7 3g x lim[ f x 7 3g x ]
4
x 1 x 1
3
x 1 x 1
limf x lim g x 1
lim f x 7 lim 3g x
152
4
x 1 x 1
3
x 1 x 1
limf x lim g x 1
lim f x 7 3limg x
4
x 1 x 1
3
x 1 x 1
1 limg x lim1
limf x lim7 3 2
3
1 2 1
1 7 6
1 3 1
2 6 4.
3.48.
32 3 2 2
x 1lim x 1 f x x 2 g x 7 2 4 1 3 7 124
3 3
2 2x 1
2x 5 g x xf x 3 3 4 77lim
21g x 3xf x 3 3 4
3.49.
2 2
x 1lim 3x 5x 4 3 1 5 1 4 3 5 4 2
2
x 2
x 3xlim
x 1
22 3 2 2
2 1 3
1 2
x 1lim x x ... x 1
-
1 1 ... 1 1
x 4
x 2 1lim
x 3
4 2 1 6 1 5
4 3 4 3 7
3.50. 33 2
f x x 3x 3x 1 x 1 1 2
x ,x 1 2
x x ... 1 2
f x f x
3
f x y x 1 y y 0 3 3x 1 y x y 1 1 3f y y 1,
y 0 3 3x 1 y x y 1 1 3f y y 1.
3
1
3
y 1 , y 0f y
y 1, y 0
.
1 3
x 0 x 0limf x lim x 1 1
3.51.
2
x 1 x 1
x 1x 5x 4lim lim
x 1
x 4
x 1 3
23 2
2x 1 x 1
x 1 x 1x x x 1lim lim
x 1 2x 1
0
2
x 3 x 3
x 3x 9lim lim
x 3
x 3
x 3 6
153
23
2x 3 x 3
x 3 x 3x 9x 27lim lim
x 3 x 2x 5x 6
22
x 3
3 3 3 9x 3x 9 9 9 9lim 27
x 2 3 2 1
3 2
3 2x 2 x 2
x 2x x x 6lim lim
x 2x 3x 6
2x x 3
x 2
2 2
2 22 x 2
x x 3 2 2 3 9lim
7x 3 2 3x 3
2
2x 1 x 1
x 2 x 1x x 2lim lim
x x x x 1 3 .
3.52.
3
2x 1 x 1
x 1x x 2lim lim
x 5x 4
2x x 2
x 1
4
3x 4
24
2x 2 x 2
x 2x 4lim lim
x 2
2
2
x 2
x 2 4
4 2
3x 1 x 1
x 1x 5x 4lim lim
x 1
2x 1 x 4
x 1
22
x x 1
23 2
3x 1 x 1
x 1x x x 1lim lim
x x 2
x 1
x 1
20
x x 2
x 0 x 0
x1 x 1 2x 1 3x 1lim lim
x
26x 11x 6
x 6
3.53.
3 3
2 2x x
xxlim lim
x
2 2x x
x
3
2x
4 2
4 3 2x 2 x 2
x 2x 2x 3x 2lim lim
x 5x 14x 80
3 2x 2x 2x 1
x 2
3 2
1
4x 3x 20x 40
3
42x 2 x 2
x 2x 8lim lim
x x 6
2x 2x 4
x 3 x 2 44
12
5
3.54.
x 1 x 1
(x 1)x 1lim lim
x 1
1 2(x x ... x 1)
x 1
x 1
1 2-
x 1
1 1 ... 1x x ... x 1
limx 1 2 2
-
1 2
1x 1 x 1
-
x 1 x x ... 1x 1 1 1 ... 1lim lim
1 1 ... 1x 1 x 1 x x 2 ... 1.
154
23 3
x 1 x 1 x 1
x x 1 x 1x 1 x x x xlim lim lim
x 1 x 1 x 1
x 1
x x 1 x 1 x 1lim
x 1
x 1
x 1lim
x x 1
x 1
x 1
lim x x 1 2
1
1x 1 x 1
x x 1 x 1x ( 1)x 1lim lim
x x x 1 x x 1 x 1
x 1
x x 1lim
x 1
1 2x x ... x 1
x 1
0
x 1
-
2 2
x 1 x 1
x x ... x x x ... x 1 1 ... 1lim lim
x 1 x 1
2
x 1
x 1 x 1 ... x 1lim
x 1
1 2
x 1
x 1 x 1 x 1 ... x 1 x x ... 1lim
x 1
1 2
x 1
x 1 1 x 1 ... x x ... 1lim
x 1
1 2
x 1
1lim 1 x 1 ... x x ... 1 1 2 ...
2 .
3.55.
2
x 3 x 3 x 3
x 3 x 3 x 3x x 9 3lim lim lim x 3 6
x 3 x 3
3.56.
x 1limf x 2 1 1
2 2 2
3 3x 1 x 1 x 1
2 x 1x f x 3 x x x 1 3lim lim lim
x 1 x 1
x 1 x 1
x 1
2
41 1
3x x 1
2
3.57.
1 2 2 1
1
x 3 x 3
x 3 x 3x ... 3 x 3x 3lim lim 3 27
x 3 x 3
13 27 1 3 4
3.58.
2 2 2 2
2
x 3
2 x 2 x 3 10x 12lim 2 5 5
x 3
2 2 2 2 2
2
x 3
x 2x 2 x 3 10x 12lim 2 5 5
x 3
2 2 2
2
x 3
x 2x 3 2 x 5x 6lim 2 5 5
x 3
155
2
2
x 3
x 3 x 1 2 x 3 x 2lim 2 5 5
x 3
x 3
x 3lim
2x 1 2 x 2
x 3 2
2 5 5
2 2
x 3lim x 1 2 x 2 2 5 5
2 2 24 2 2 5 5 2 5 3 0 3
1
2 .
2 2 2
2
x 2
2 x 4 6x 4lim 3 4
x 2
2 2 2 2
2
x 2
x 2x 4 6x 4lim 3 4
x 2
2 2 2
2
x 2
x 4 2 x 3x 2lim 3 4
x 2
2
2
x 2
x 2 x 2 2 x 1 x 2lim 3 4
x 2
x 2
x 2lim
2x 2 2 x 1
x 2 2
3 4
2 2
x 2lim x 2 2 x 1 3 4 2 2
4 2 3 4
22 0 2 1
3.59. 33 2
f x 3f x 3f x x 9 f x 1 x 8 .
1 2
x ,x 1 2 1 2
f x f x ... x x .
3 3
f x y y 1 x 8 x y 1 8 ,
3 21f x x 1 8 x 3 x 1 2 x 1 4 .
21
2x 3 x 3
x 3 x 1 2 x 1 4f xlim lim 12
x 3 x 2x 5x 6
3.60.
2x 7 x 7
2 x 3 2 x 32 x 3lim lim
x 49 x 7 x 7 2 x 3
x 7
x 7lim
x 7
1
56x 7 2 x 3
x 1 x 1 x 1
2 x 3 2 x 3 x 12 x 3lim lim lim
x 1 x 1 2 x 3 x 1
1
42 x 3
h 0 h 0 h 0
1 h 1 h 1 h 1 h1 h 1 h 2hlim lim lim
h h 1 h 1 h h
1
1 h 1 h
156
x 0 x 0
2 x 2 x 2 x 2 x2 x 2 xlim lim
x x 2 x 2 x
x 0
2 xlim
x
2
22 x 2 x
x 1 x 1
2x 3 1 2x 3 1 5 x 22x 3 1lim lim
5 x 2 5 x 2 5 x 2 2x 3 1
x 1
2 x 1lim
5 x 2
x 1
4
2x 3 1
x 3
x 3lim
x 2 1
x 3
x 3 x 3 x 2 1lim
x 2 1 x 2 1 x 3
x 3
x 3 x 2 1lim
x 2 1 x 3
x 3
x 3lim
x 2 1
x 3
x 3
x 2 1 2 3lim
3x 3 2 3x 3
x 0 x 0 x 0
x 9 3 x 9 3 9 x 3 xx 9 3lim lim lim
9 x 3 9 x 3 9 x 3 x 9 3
9 x 3
x
1
x 9 3
x 1 x 1
x 8 3 x 8 3 x 7 8x 8 3lim lim
x 7 8 x 7 8 x 7 8 x 8 3
x 1
x 1lim
x 7 8
x 1
2 2
3x 8 3
x 2 x 2
x x 2 x x 2 4x 1 3x x 2lim lim
4x 1 3 4x 1 3 4x 1 3 x x 2
x 2
x 2lim
x 1 4x 1 3
4 x 2
9
8x x 2
3.61.
2 2
2x 3
x 2x 6 x 2x 6lim
x 4x 3
2 2 2 2
x 3 2 2
x 2x 6 x 2x 6 x 2x 6 x 2x 6lim
x 1 x 3 x 2x 6 x 2x 6
x 3
4 x 3lim
x 1 x 3
2 2
1
3x 2x 6 x 2x 6
157
x 1 x 1
x 8 8x 1 x 8 8x 1 5 x 7x 3x 8 8x 1lim lim
5 x 7x 3 5 x 7x 3 5 x 7x 3 x 8 8x 1
x 1
7 x 1lim
5 x 7x 3
8 x 1
7
12x 8 8x 1
3 33 3
23 3x 2 x 2 x 2
x 8 x 8x 8 x 8
lim lim limx 4 x 2 x 2 x 8 x 2 x 2 x 8
x 2
x 2lim
2x 2x 4
x 2
3
3 2
8x 2 x 8
3.62.
2
2x 1 x 1 2 2
f x 7 3 f x 7 3 2 2f x 4f x 7 3lim lim
2 2f x 4 2 2f x 4 2 2f x 4 f x 7 3
x 1
f x 2lim
22 2f x 4
2 f x 2
1
12f x 2 f x 7 3
3.63.
23 3 3
3
23 3
x 3 x 3
x 5 2 x 5 2 x 5 4x 5 2
lim limx 3
x 3 x 5 2 x 5 4
x 3
x 3lim
x 3
23 3
1
12x 5 2 x 5 4
3 3
x 5
x 2 3lim
x 5
2 23 3 33 3 3
2 23 3 3 3
x 5
x 2 3 x 2 x 2 3 3
lim
x 5 x 2 x 2 3 3
2 23 3 3 3
x 5
x 5lim
x 5 x 2 x 2 3 3
2 2
3 3 3 3x 5
1lim
x 2 x 2 3 3
3 3
9.
23 3
233 3 3
x 1 x 1
2 x 1 2x 6 2 2x 6 42x 2
lim lim2x 6 2 2x 6 2 2x 6 2 2x 6 4
x 1
2 x 1
lim
23 32x 6 2 2x 6 4
2 x 112
158
2 23 3 3 3 3 3
3 3
2 23 3 3 3
h 0 h 0
x x h x x x h x hx x h
lim limh
h x x x h x h
h 0
hlim
h
3
2 23 3 3 3
x
3xx x x h x h
2
3 3x 2
x 2x 8lim
x 2 3x 2
2 22 3 33 3
x 2 2 23 33 33 3
x 2x 8 x 2 x 2 3x 2 3x 2
lim
x 2 3x 2 x 2 x 2 3x 2 3x 2
2 22 3 33 3
3 3x 2 3 3
x 2x 8 x 2 x 2 3x 2 3x 2
lim
x 2 3x 2
x 2
x 2
lim
2 23 33 3x 4 x 2 x 2 3x 2 3x 2
2 x 2 318 2
23 3 3
3
2x 2 x 23 3
x 6 2 x 6 2 x 6 4 x 7 3x 6 2
lim limx 7 3 x 7 3 x 7 3 x 6 2 x 6 4
x 2
x 2lim
x 7 3
x 2
23 3
1
2x 6 2 x 6 4
3.64.
x 4 x 4
x 4x 4 x 4lim lim
x 4 x 2
x 4 x 4
x 4
0
x 2
2
x 2 x 2
x 2 x 2 x 2x 2 x 4lim lim
x 2 x 2
x 2
x 2 x 2 x 2lim
x 2
x 2 x 2
x 2 1 x 2lim lim 1 x 2 1 2 1
x 2
x u , u 0 2x u x 4 u 2 .
:
2 3
2 2x 4 u 2 u 2
x x 8 u u 8 u 8lim lim lim
u u 6 u u 6x x 6
u 2
u 2lim
2u 2u 4
u 3 u 2
2
u 2
u 2u 4 12lim
u 3 5.
159
2 2 2 2 2 2x x
x x x xlim lim
x x x
x
x x xlim
x x x x
x
2x
(x ) x 1lim
xx x x
x
xlim
x
x
1
xx x
x
x 1 2lim
2xx x
2
x 1 x 1
5 x x 1x 5 x 5 x x xlim lim
x 10 x 10 x x 1
x x 1
10 x x 1 x 1
3
2
3.65.
3 2x 4 x 4
x 4x 4 x 4lim lim
x 2 x 2 x 4 x 5
x 4 x 2
x 4
3 2
x 2 x 4 x 5
32
29
x 4 x 4
x x 4x x 4lim lim
x 2 x 3
x 2
x 4
16
x 3
x 1 x 1
x 5 3 x 3 x 5 3 x 3 x 2 x 8x 5 3 x 3lim lim
x 2 x 8 x 2 x 8 x 2 x 8 x 5 3 x 3
x 1
x 1lim
x 2 x 2 x 8
x 1
3
10x 4 x 5 3 x 3
3.66. x 3 3 3x 24 27 3 x 13 16 4
1 4 3
3
x 3
x 24 x 13 1lim
x 3
3
x 3
x 24 x 13 4 3lim
x 3
3
x 3
x 24 3 x 13 4lim
x 3 x 3
333
2x 33 3
x 24 3x 13 16
limx 3 x 13 4x 3 x 24 3 x 24 9
160
x 3
x 3lim
x 3
23 3
x 3
x 24 3 x 24 9 x 3
x 13 4
2x 3 3 3
1 1lim
x 13 4x 24 3 x 24 9
1 1 19
27 8 216.
3 3
2x 5 x 5
x 3 x 4 1 x 3 2 x 4 3lim lim
x 3 x 5 x 3 x 5x 8x 15
23 3 3
2x 53 3
x 3 2 x 3 2 x 3 2 x 4 3 x 4 3lim
x 3 x 5 x 4 3x 3 x 5 x 3 2 x 3 2
x 5
x 5lim
x 3 x 5
23 3
x 5
x 3 2 x 3 4 x 3 x 5
1
24x 4 3
x 2 x 2
2 x 2 3x 2 6 x x 2 3x 2 x 2 6 xlim lim
x 2 x 2 x 2
x 2
x 2 3x 2 x 2 3x 2 x 2 6 x x 2 6 xlim
x 2 x 2 3x 2 x 2 x 2 6 x
x 2
2 x 2lim
x 2
2 x 2
x 2 3x 2 x 2
0x 2 6 x
3 3 3 3
x 1 x 1
2x 1 x 7 3x 2 2x 1 1 x 7 2 3x 2 1lim lim
x 1 x 1 x 1 x 1
x 1
2 x 1lim
x 1
23 3
x 1
2x 1 2x 1 1 x 1
23 3
3 x 1
x 7 2 x 7 4 x 1
3x 2 1
25
12
3.67.
62
3 23 x u
2 12x 1 u 1 u 1
u u 1x x u ulim lim lim
x 1 u 1 u 1
2 3 6
1
12u u 1 u 1 u 1
)
12 4 33 4 x u
12u 1x 1 u 1 u 1
u 1x x 2 u u 2lim lim lim
x 1 u 1
3 2u 2u 2u 2
u 1
2 3 6
7
12u u 1 u 1 u 1
20 54 x u
45x 1 u 1 u 1 u 1
u 1x 1 u 1lim lim lim
u 1x 1
4 3 2u u u u 1
u 1
2
5
4u 1 u 1
161
x 1 2,3,6 6 .
: 6 x 1 y 2 23 6x 1 x 1 y
3 36x 1 x 1 y
x 0 y 1.
3 23
36x 0 x 1
x 1 2 x 1 1 y 2y 1lim lim
y y 2x 1 x 1 2
x 1
y 1lim
2y y 1
y 1
2
22 x 1
y y 1 1lim
4y y 2y y 2
6 36 3x 1 u
23 6 u 1 u 1 u 1x 0
u u 13x 1 3x 1 u ulim lim lim
u u3x 1 3x 1
u 1
u u 1 2
63 6 4x 1 u
x 0 u 1
4x 1 3 4x 1 2lim
4x 1 1
2
3u 1 u 1
u 1u 3u 2lim lim
u 1
u 2
u 1
2
1
3u u 1
3.68.
2 4 2x u
3 2u 1x 1 u 1 u 1
u 1x 4x 3 u 4u 3lim lim lim
u 3u 2ux x 3x 2 x
2u 1 u 3
u u 1
4
u 2
2 x x 5 7 2 x x 5 4 3
x 4 x 4
2 x 4 x 5 3
x 4 x 4
2 x 4 2 x 4 x 5 3 x 5 3
x 4 2 x 4 x 4 x 5 3
4 x 4 x 4 4 1
2 x 4 x 5 3x 4 2 x 4 x 4 x 5 3.
x 4
2 x x 5 7 4 1 2lim
x 4 8 6 3
x 4
x 4 3lim
22 x x 5 7.
2 4 2x u
3 2x 4 u 2 u 2 u 2
u 2x 6x 8 u 6u 8lim lim lim
u u 6ux x x 6 x
2u 2 u 2
u u 3 u 2
4
5
x 6 x 6 0
2x 6 x 6 x 6
2 2
2x 6 x 6 x 6
x 6 x 6x 6lim lim lim
x 2 x 2 4x 2 2 x 2 x 2 4 x 2
x 2 2 x 2x 2 2 x 2
2
x 6 x 6
x 6 x 2 2 x 2 x 6 x 2 2 x 2lim lim 0
x 2 x 6 x 2.
162
3.69.
2 23 3 3 3
x 2 x 2
3x 4 3x 2 8 3x 4 3x 2 4 3x 2 4 3x 2 8lim lim
x 2 x 2
2 3
3
x 2
3x 4 4 3x 2 2 3 3lim 3x 2 4 2 4 4
x 2 x 2 2 12
2 22 2
x 2 x 2 x 22 2
3x 4 4 3x 4 43x 4 4 3x 4 16
lim lim limx 2 x 2 3x 4 4 x 2 3x 4 4
x 2
3 x 2lim
x 2
x 2
2
12 3
8 23x 4 4
23 3 3
3
2x 2 x 23 3
3x 2 2 3x 2 2 3x 2 43x 2 2
lim limx 2
x 2 3x 2 2 3x 2 4
333
2x 23 3
3x 2 2lim
x 2 3x 2 2 3x 2 4
2x 23 3
3x 2 8lim
x 2 3x 2 2 3x 2 4
x 2
3 x 2lim
x 2
23 3
3 1
4 4 4 43x 2 2 3x 2 4
x 4 x 4
x 4 x 4lim lim
x 2 9x x 2 9x x 2 9x
x 2 9x
2x 4 2
x 4lim
x 2 9x
x 2 9x
2x 4
x 4 x 2 9xlim
x 4x 4 9x
2x 4 x 4
x 4 x 2 9x x 4 x 2 9xlim lim
x 4 x 1x 5x 4
2
x 4
x 4lim
x 2 9x
x 4
x 4
x 4 x 2 9xlim 0
x 1x 1
3.70. f 1 x 1 x 0 x 0 .
11 x 0 x
11 x 0 x .
1
1 1 1 11 1 0
1 1A , ,0 0,
x 1 1
1 x + +
1 x +
+ +
163
x 0 x 0
1 x 1 x 1lim f x lim
x
x 0
1 x 1 x 1 1 x 1 x 1lim
x 1 x 1 x 1
22
x 0
1 x 1 x 1lim
x 1 x 1 x 1
2
x 0 x 0
x x1 x x x 1lim lim
2x 1 x 1 x 1 x 1 x 1 x 1.
3.71.
2 2
x 1 x 1
lim f x lim x x 1 1 1 1 1,
x 1 x 1
lim f x lim 2x 1 2 1 1 1
x 1 x 1
lim f x lim f x 1,
x 1lim f x 1
x 0 x 0 x 0
2 x 1 1 x 1 12 x 1 2lim f x lim lim
x x x 1 1
x 0
2 x 1 1lim
x x 1 1
x 0 x 0
2x 2 2lim lim 1
1 1x 1 1x x 1 1
x 0 x 0
lim f x lim 6x 1 1
x 0 x 0
lim f x lim f x 1
x 0lim f x 1.
3.72. x 3 x 3lim f x
2
x 3 x 3 x 3
x 2 x 3x 5x 6lim lim lim x 2 1
x 3 x 3
x 1 :
2
x 0 x 0
x 1lim g x lim 1
x 1.
3.73. :
2 2 2
x 2 x 2
lim f x lim x 3 x 5 4 6 5
2 2 2 2
x 2 x 2
lim f x lim x x 1 4 2 1 2 5 .
:
2 2
x 2 x 2
lim f x lim f x 4 6 5 2 5
22 6 0 2 3 0 0 3 .
3.74. f
C A( 2, 8) ,
f 2 8 2 2 3 8 4 3 8 1
x 1
limf x
x 1 x 1
lim f x lim f x .
x 1 x 1
lim f x lim 2 x 3 2 3
2
x 1 x 1 x 1
x 1x 1lim f x lim lim
x 1
x 1 ( x 1)
x 1
x 1
lim x 1 x 1 4 .
2 3 4 2
1 , 2
4 3 8 2
02 3 4
164
3.75. x 3
lim f x
x 3 x 3
lim f x lim f x .
2 2 2
x 3 x 3
lim f x lim x x 1 3 9 4 3
2 2 2 2
x 3 x 3
lim f x lim 2 3 x 5 1 3 2 3 5 1
2 2 26 9 5 1 6 4 1
2 2 2 29 4 3 6 4 1 9 6 4 4 2 0
2 22 2
9 6 1 4 4 1 0 3 1 2 1 0
2
3 1 0 2
2 1 0
3 1 0 1
2 1 03
1
2.
3.76.
x 1 x 1
lim f x lim f x 2 1 ,
x 2 x 2
lim g x lim g x 2 4 2
1 , 2 2 , 0
3.77.
x 1 x 1
lim f x lim f x 6 1 ,
x 3 x 3
lim f x lim f x 3 2 2
1 , 2 4 , 2
3.78.
x 2 x 2
1lim g x lim g x 2
2 , 2
3.79.
2 2
x 3 x 3 x 3 x 3
x x 6 x 3 x 2x x 6 x 2lim lim lim lim 5
x 4 1 x 4 1 x 3 1
x 1
x 2 1lim
x 1
x 1 x 1
x 2 1 x 1lim lim
x 1 x 1
x 1
x 1lim 1
x 1
2
x 3
x 1 x 4 9lim
x 3
x 3
x 1 x 2 x 2 9lim
x 3 1
1 :
2
x 3
x 1 x 4 9lim
x 3
2 2
x 3 x 3
x 1 x 4 9 x x 12lim lim
x 3 x 3
x 3
x 4 x 3lim
x 3
x 3lim x 4 7
x 0 x 0 x 0
x 2 x 2 x 2 x 2 2xlim lim lim 2
x x x
22
x 1 x 1
x 1x 2x 1lim lim
x 1 x 1 0
x 2 1 2 3
x 1 + +
x 2 +
x 2 + + +
165
2 2 2
2 22x 1 x 1 x 1
x 7x 10 x 5 x 7x 10 x 5 x 6x 5lim lim lim
x 2x x x xx 2x x
x 1
x 1lim
x 5
x x 1 4
3.80.
2 2
2 2x 0 x 0 x 0
5x 4 x x5x 4xlim lim lim
x 2 x x 2x
5x 4
x
2
x 2
2 2
2 2x 0 x 0 x 0
5x 4 x x5x 4xlim lim lim
x 2 x x 2x
5x 4
x
2
x 2
2
2x 0
5x 4 xlim 2
x 2 x
2 2x 1 x 1
x 1 x 2 1 x 1 x 2 1lim lim 0
x x x x
2x 1 x 1
2 x 1x 1 x 2 1lim lim
x x x x 1 2
2x 1
x 1 x 2 1lim
x x.
2 2
22x 2 x 2 x 2
x 2 xx x 2 2 x x x 2 2 xlim lim lim
4 x4 x 2 x
1
22 x
2 2
22x 2 x 2 x 2
x x 2x x 2 2 x x x 2 2 xlim lim lim
x 44 x x 2
1
2x 2,
2
2x 2
x x 2 2 x 1lim
24 x
2 2
2
x 5x 6 9 xf x
x 3 x x 3, x 3
2 2
2x 3 x 3
x 5x 6 9 xlim f x lim
x 3x x 3
2 2
2x 3
x 5x 6 9 xlim
x 3x x 3
2x 3
5x 15lim
x 4x 3
x 3
5 x 3lim
x 1 x 3
x 3
5 5lim
x 1 2
2 2
2x 3 x 3
x 5x 6 9 xlim f x lim
x 3x x 3
2x 3 x 3
5 x 35x 15lim lim
x 3 x 1x 2x 3
x 3
5x 5lim .
x 1 4
x 3 x 3
lim f x lim f x x 3
lim f x .
x 3 0 2 3
x
2x 5x 6
29 x
x 3
166
22
x 4 x 4 x 4
x 4 x 4x 8x 16lim lim lim
x 4 x 4 x 4
x 4x 4
x 4 x 4lim lim 1
x 4 x 4,
x 4 x 4
x 4 x 4lim lim 1
x 4 x 4,
2
x 4
x 8x 16lim
x 4
2
x 4 x 4
x 1 x 4x 5x 4lim lim
x 2 2 x 4 3
2
x 4 x 4
x 1 x 4x 5x 4lim lim
x 2 2 x 4 3
2
x 4
x 5x 4lim
x 2 2.
3.81.
2 22 2
x 3 x 3
x 4x 3 x 2x 3 x 1 2x 4x 3 x 2x 3lim lim
x 1 2 x 1 2 x 1 2
2 2
x 3
x 4x 3 x 2x 3lim 8
x 1 2,
2 2
x 3
x 4x 3 x 2x 3lim 24
x 1 2,
22
x 2 x 2
x 2 x 2 3x 2x 4x 4lim lim
x 2 3x 2 x 2 3x 2 x 2 3x 2
x 2
x 2 x 2 3x 2lim
2 x 2,
x 2 x 2
x 2 x 2 3x 2 x 2lim lim
2 x 2
x 2 3x 2
2 x 2 2 ,
x 2 x 2
x 2 x 2 3x 2 x 2lim lim
2 x 2
x 2 3x 2
2 x 2 2 ,
2 2 2
22x 2 x 2
x 5 11 x x 5 11 x x 5 11 x
lim limx 4 x 2 x 2 x 5 11 x
2
2x 2
x x 6lim
x 2 x 2 x 5 11 x
2x 2
x 3 x 2lim
x 2 x 2 x 5 11 x
x 2
x 2lim
x 3
x 2
2
5
24x 2 x 5 11 x
167
2 3
2x 1 x 1 x 1
x 11 x x x 2 3 1 x x 2x 3lim lim lim
x 1 x 1x 1
2x x 2
x 1
2
x 1
2 3
2x 1 x 1 x 1
x 11 x x x 2 3 1 x x 2x 3lim lim lim
x 1 x 1x 1
2x x 4
x 1
3
x 1
3.82.
2
x 3 x 3 x 3
x 3 x 3 x 3 x 3 x 3 x 3lim f x lim lim
x 3 x 2 x 3 x 2,
x 3 x 3
x 3lim f x lim
x 3
x 3
x 3
5
x 2,
x 3 x 3
x 3lim f x lim
x 3
x 3
x 3
7
x 2
3.83.
2 2 2 2
x x 2 2
x 2 x 2 x 2 x 2lim f x lim
x x 2 x 2
2
x 2 2
xlim
x x 2 x 2
0
x x
xlim f x lim 1
x
0
2
x x
xlim f x lim
x
2 2
0x 2 x 2
0
2
x x 2 2
xlim f x lim 0
x x 2 x 2
3.84.
x 3
1 x 3lim
x 3
x 3
x 3 7 ,
x 3
1 x 3lim
x 3
x 3
x 3 5
7 5 1
3.85.
x x2 2
2x x 1lim g x lim
2x,
x x2 2
2xlim g x lim
x 1
2x
22
2,
x x2 2
2xlim g x lim
x 1
2x
22
2,
2 2
22 22 0 0
2 2 2
168
3.86.
2 22 22 2
x x 2 2
x xxlim lim
xx x
2 2
x
xlim
2
2
x2 22 2
xlim
x x x x
.
0 x x 0
2
x
xlim
x
x2 22 2
xlim 0
x x
0 x x 0
2 22 2
x x
x xlim lim
x x
2 2 2
x x xlim lim lim 1
0
2 2 2
x x x
x xxlim lim lim 1
x x x.
3.87. 2 3 23x 5x 7 f x 4 x 6x 4x 2
2 3 23x 5x 7 4 f x x 6x 4x 2 4
2 3 23x 5x 3 f x x 6x 4x 2
:
2
x 1lim 3x 5x 3 3 5 3 5
3 2
x 1lim x 6x 4x 2 1 6 4 2 5
x 1limf x 5 .
3.88.
x 4 x 4 x 4
x 2 x 2x 2 x 4lim lim lim
x 4 x 4 x 2 x 4
x 4
1 1lim
4x 2x 2
2x 4 x 4
x 44 xlim lim
x 12x 32 x 4
x 4
1 1lim
x 8 4x 8.
x 4
1lim f x
4.
3.89. x 0
4 8 4 8
3 7 3 7g x g xx x x x
x x x xx x x x
3 7
x 0
lim x x 0 ,
3 7
x 0
lim x x 0
x 0
g xlim 0
x 1 .
169
x 0
4 8 4 8
3 7 3 7g x g xx x x x
x x x xx x x x
3 7
x 0
lim x x 0 ,
3 7
x 0
lim x x 0
x 0
g xlim 0
x 2 .
1 , 2
x 0
g xlim 0
x.
3.90. : 22
x 1 x 1 23
x 1 f x (x 1) x 1 .
x 1
23x 1 f x (x 1) x 1
x 1 x 1
2x 1 f x x 1 x x 1
x 1x 1
2f x x x 1 x 1
2x 1 f x x x 1 x 1 2 2
x x 1 x 1 f x x x 1 x 1
2
x 1lim x x 1 x 1 3
2
x 1lim x x 1 x 1 3 .
x 1limf x 3 .
3.91.
x 2lim2 2x 2 4 4
x 2lim x 2 4
x 2lim f x 4 .
2 2x 4 f x 4 x 2 .
x 2
2 2x 2f x 4 f x 42 2x 4 x 21
x 2 x 2 x 2 x 2 x 2.
x 2 x 2 x 2
2 2x 2 4 x 22 2x 4lim lim lim
x 2 x 2 2x 2 x 2
41
42x 2
x 2
f x 4lim 1
x 2 1 .
x 2
f x 42 2x 41
x 2 x 2
x 2 x 2 x 2
2 2x 42 2x 4 4lim lim lim 1
x 2 2x 2x 2 2x 2
x 2
f x 4lim 1
x 2 2
1 , 2
x 2x 2 x 2
f x 4 f x 4 f x 4lim lim lim 1
x 2 x 2 x 2.
2x 2 x 2
f x 5 3 f x 5 9lim lim
x 4 x 2 x 2 f x 5 3
x 2
f x 4 1 1 1lim 1
x 2 4 3 3 24x 2 f x 5 3
170
x 2lim f x 4
x 2lim f x 3 1 0 f x 3 0 .
2x 2 x 2 x 2
f x 3 1 f x 4 f x 4 1 1lim lim lim 1 1
x 2 x 3 x 2 x 3 1x 5x 6.
3.92.
2
x 1lim x 4x 5 ,
2
x 1lim 3x 2 5
x 1limf x 5
2 2x 4x 5 f x 5 3x 3 x 1
x 5 x 1
x 1
x 1f x 53
x 1
x 1
x 1
x 1
f x 5lim 6
x 1x 1
x 1
f x 5lim 6
x 1
x 1limf x 5 f x 1 0 x 1,
x 1 x 1 x 1
f x 5 x 3 2f x 1 4 f x 5lim lim lim x 3 2 24
x 1x 3 2 x 3 2 x 3 2
x 1
f x 4 3 f x 4 3 2f x 6 2lim
2f x 6 2 2f x 6 2 f x 4 3
x 1
f x 5lim
2f x 6 2
2 f x 5
1
3f x 4 3
3.93. 4g x 0 3f x 4g x 3f x 3f x 0
0x x
lim 4g x 3f x 0 ,
0 0x x x x
lim 3f x 0 lim f x 0 .
h x 3f x
4g x 3f x h x g x4
0 0x x x x
h x 3f xlim g x lim 0
4
3.94. 2 2f x f x f x f x f x f x .
0
2
x xlim f x 0
0
2
x xlim f x 0
0
2
x xlim f x 0
0x x
lim f x 0 .
3.95.
2 2
x 2 x 2lim 4f x f x 4 lim 4f x f x 4 0
2
x 2lim f x 2 0
2
x 2lim f x 2 0
2 2 2 2
f x 2 f x 2 f x 2 f x 2 2 f x f x 2 2
x 2lim f x 2
3.96. 22 2 2
f x 2f x x 0 0 f x 1 x
2
x 0lim f x 1 0
2 2 2 2
f x 1 f x 1 f x 1 f x 1 1 f x f x 1 1
x 0lim f x 1
171
3.97. 22 2 2
f x 4f x x 4 f x 2 x .
2
0 f x 2
2
x 0limx 0
2
x 0lim f x 2 0 .
2 2
f x 2 f x 2 f x 2
x 0 x 0lim f x 2 0 limf x 2
3.98. 2 2 20 f x f x g x
0
2
xlim f x 0
0x
lim f x 0 g.
3.99.
2 22 2
x 1 x 1lim f x g x 6f x 8g x 25 lim f x 3 g x 4 0
2 2 2
0 f x 3 f x 3 g x 4
2
x 1lim f x 3 0
x 1limf x 3
g
3.100.
0 0
3 3
x x x xlim g x g x 1 lim g x g x 1 0 1 .
3 3g x g x 1 g x 1 g x 1
2g x 1 g x g x 1 g x 1
2g x 1 g x g x 1 .
2g x g x 0 1 4 0
3
2
g x g x 1g x 1
g x g x 1.
2g x g x 1 1 x .
3 33
2 2
g x g x 1 g x g x 1g x g x 1g x 1
1g x g x 1 g x g x 1
3 3g x g x 1 g x 1 g x g x 1
3 31 g x g x 1 g x 1 g x g x 1 .
0
3
x xlim 1 g x g x 1 1,
0
3
x xlim 1 g x g x 1 1.
0x x
lim g x 1.
3.101.
5
4
x 3f x f x 3 x f x
f x 1,
4 4
x 3x 3f x x 3
f x 1 f x 1
4
4
1f x 1 1 1
f x 1 x 3 f x x 3
x 3lim f x 0
3.102. 2
2 2 2 2 2f x f x g x 16f x 9g x 4f x 3g x 24f x g x x .
20 f x x
0 0
2
x x x xlim x lim 4f x 3g x 24f x g x 0 24 0 0 .
172
0 0
2
x x x xlim f x 0 lim f x 0 .
h x 4f x
h x 4f x 3g x g x3
0 0x x x x
h x 4f x 0 4 0lim g x lim 0
3 3.
3.103. 2 2 2
h x g x h x g x 2h x g x
2 2 21h x g x h x g x h x g x
2.
0x xlim h x g x 0 .
2 2 2
h x g x h x g x 2h x g x
0 0 0
2 2 2
x x x x x xlim h x g x lim h x g x 2 lim h x g x 0 .
2 2 20 h x h x g x
0
2
x xlim h x 0
0x xlim h x 0 .
0
2 2 2
x x0 g x h x g x ... lim g x 0 .
3.104. u
3x u x3
. x 0 u 0 ,
:
x 0 u 0 u 0
3x u 3 ulim lim lim
ux u
3
u 0
u3lim 3 1 3
u
x 0
2xlim
x
x 0 x 0
2x
2 2xlim lim 2x x 1
x x
x 0
xlim
x
x 0 x 0
x
xxlim limx x x
x 0
x 1lim 1
x x
x 0
1 x 1lim
x
x 0
1 x 1 1 x 1lim
x 1 x 1
x 0
1 x 1lim
x 1 x 1
x 0
x 1 1 1lim 1
x 1 1 21 x 1
x 0
3x xlim
x
x 0 x 0 x 0
3x x 3x xlim lim lim 3 1 2
x x x x
2x 0 x 0
1 2x 1 2x 1lim lim 2 0
2x x 1x x
2
2 2 2x 0 x 0 x 0
2x 1 2x 12x 1 2xlim lim lim
x x 2x 1 x 2x 1
x 0
2x 1lim 4 2
2x 2x 1
173
2 x 0x 0
4 x x 4lim lim 2
x x 2x 2x
x 0 x 0 x 0 x 0
2x 2x 2x 2xlim lim lim lim 2x 0 1
2x2x 2x
2x
.
3.105.
x 0 x 0
1 x 1 x 1 x 1 x1 x 1 xlim lim
x x 1 x 1 x
x 0
x 1lim 2 1
x 1 x 1 x
3x 0 x 0
x 2x 3x x 2x 3xlim lim
x x xx
x 0
x 2x 3xlim 2 3 1 2 3 6
x 2x 3x.
x 0 x 0
3x3
3x 33xlim lim4x4x 4
44x
x 0 x 0
5x4 x 5
4x x 5x 5xlim lim 32xx 2x
1 22x
x 0 x 0
x3
3x x 1xlim limx3x x 2
3x
x 0 x 0 x 0
3x x 9 33x 3xlim lim lim 3 x 9 3 18
3xx 9 3 x 9 3 x 9 3
3.106.
x 3 u
2x 3 x 3 u 0 u 0
x 3 x 3 u 1lim lim lim 1
x 3 x 4 u u 1x 7x 12
2 2x 0 x 0
x x xlim lim
x4x 3x 4x 3x
x 0
x x 1lim
x x 4x 3
1 11 1
3 3.
x 0 x 0
2 x 2 x 2 x 2 x2 x 2 xlim lim
x x 2 x 2 x
x 0
x 1 2lim 2
x 22 x 2 x
174
2
x 0 x 0
x 1 2x
x x 1 2x x xlim limxx x
x
2
2
x 0
x 1 2x 14
x x 1 2x4xlim 3x
x
3.107.
2 2
x 0 x 0 x 0
xx x x x xlim lim lim
x 1x x x x,
2 2
x 0 x 0 x 0
xx x x x xlim lim lim
x 1x x x x,
2 1, 0
3.108. f x y ... 1 3 2f x x 3x 3x 1,
1 3 2 2
x 0 x 0 x 0
f (x) 1 x 3x 3x 1 1 x 3x 3lim lim lim 3
xx x
x
3.109.
2 4
2x 0 x 0 x 0
x x
x x x xlim f x lim lim
x x
21 x
x
2
x 1,
2
x 0 x 0
lim f x lim x 2
3.110.
2 2
2 2 2x 0 x 0
2 2
xf 4x f x 3x f 4x f x 3x
x x x x xlim lim4x x x
4x x x
2x 0
f x f x 3x4 3
4x x 3xlim
x4
x
.
4x u
x 0 u 0 u 0
f 4x f ulim lim 3
4x u,
x
x 0 0 0
f x flim lim 3
x
x 0
3xlim 1
3x.
2 2x 0
f x f x 3x4 3
4 3 3 3 14x x 3xlim 74 1x
4x
.
3.111.
x 0f x
g x f x xg xx
2 2 2
3 3 2 3 3 2x 0 x 0
f x x f x x 3 xg x x x g x x 3lim lim
x 3x x x 3x x
2
2
2
x 0
xg x g x x 3
xlim 2x
x 3xx
175
3.112.
x 0 x 0
x x xlim lim 1 0
x x
2 6
x 0lim x x 0
x 0lim f x 0
3.113. x 0
22x2 1 f x x 3
2x
x 0
2xlim 2 1 3
2x,
2
x 0lim x 3 3 ,
x 0lim f x 3
3.114. 2 2 2 2
f x x x g x 2 xf x xg x x x
2 2
f x x g x x x x
x 0lim x x 0
2 2
x 0lim f x x g x x 0 .
2 2 2
0 f x x f x x g x x
2 2
x 0lim f x x g x x 0 ,
2
x 0lim f x x 0 .
2 2
f x x f x x f x x
x 0lim f x x 0 .
f x x h x f x h x x
x 0 x 0limf x lim h x x 0 .
x 0limg x 0 .
3.115. 2 2 2x 4x 4xf x f x x 4x 4xf x
2
x f x 2x x 1
2
x 0lim f x 2x 0 .
2 2 2 2
f x 2x f x 2x f x 2x f x 2x 2x f x f x 2x 2x
x 0lim f x 0
3.116. 22 2 2 2 2
f x 2f x x 0 f x 2f x 1 1 x f x 1 x .
2
x 0lim x 0
2
x 0lim f x 1 0 .
2 2 2 2
f x 1 f x 1 f x 1 1 f x 1 f x f x 1 1.
x 0 x 0lim f x 1 0 limf x 1
3.117.
0
0
x x h0 0 0 0 0 0
x x h 0 h 0 h 00 0 0
f x f x f x h f x f x 2h f h 2x f xlim lim lim
x x x h x h
0 0
0 0h 0 h 0
f x 2h 1 f h 2x f h2h 1lim lim 2f x 2x
h 2h h
0 0 0
f x 0 2 2x 2 2x
176
3.118. 3f x f x 2x 3f x 2x f x 3f x 2x f x ,
f x f x 3f x 2x f x 1 .
3f x 2x 3 f x 2 x 2 .
1 2 3f x 2x 3f x 2x f x
3 f x 2 x f x 2 f x 2 x f x x x f x x 3 .
x 0lim x 0
x 0lim x 0
x 0lim f x 0 .
3 x 0 0 f 0 0 f 0 0 .
x 0 f x 0
f x f x f xx x 3 1 13 2
f xf x f x f x 2 2 f x x 3 1
2 2 f x
,
x 0 x 0
f x 1 1lim lim 1
3 1f xx 3 1 12 22 2 f x
.
3.119. x u x u x u 0
x u 0 u 0
ux x u u ulim lim lim
2 2 2 2 2 2 2
u 0 u 0
u
u2
u u 1 0u 2lim lim
u2 2 1
2
u
2
3.120. 5
f x xx
, A *
5 5
f x x x x 1 xx x
x f x x
x 0lim x 0
x 0lim x 0 ,
x 0lim f x 0
2 2 22x 5 2x 5x 3x x 3x x 3x
x 3 x 3
2 2 22x 5x 3x x 3x x 3x
x 3.
2
x 3
2x 5lim x 3x 0
x 3
177
2 2 2
x 9 3 3 x 9 3 3 x 9 3
x x x x x
2 2 2x 9 3 x 9 3 3 x 9 3
x x x x.
2 22 2
x 0 x 0 x 02
x 9 3 x 9 3x 9 3 x
lim lim limx x x 9 3 x
2
0x 9 3
,
2
x 0
x 9 3 3lim 0
x x
2 2 24x 1 4x 1x 2x x 2x x 2x
x 2 x 2
2 2 24x 1x 2x x 2x x 2x
x 2
2
x 2
4x 1lim x 2x 0
x 2
2 2 2 2
2 2
2x 3 2x 3x 1 x 3x 2 x 1 x 3x 2
x 1 x 1x 1 x 1
2 2
2
2x 3x 1 x 3x 2
x 1 x 1 2 2
x 1 x 3x 2
2 2 2 2 2 2
2
2x 3x 1 x 3x 2 x 1 x 3x 2 x 1 x 3x 2
x 1 x 1
2 2
x 1lim x 1 x 3x 2 0 ,
2 2
2x 1
2x 3lim x 1 x 3x 2 0
x 1 x 1
2
2
x 3f x x 2x
xx 1, A *
2 2 2
2 2 2
x 3 x 3 x 3f x x 2x x 2x x 2x
x x xx 1 x 1 x 1,
:
2 2
2
x 3f x x 2x x 1 2x 1
xx 1
2x 2x 2 2
x 2x f x x 2x .
2
x 0lim x 2x 0
2
x 0lim x 2x 0 ,
x 0lim f x 0 .
3.121.
1821 1821 1821x 2 x 2x x x x x x
x x
1821 1821 1821x 2
x x x x x xx
.
1821
x 0
x 2lim x x 0
x
178
200 200
2 2
2 2x 1 x 1 x 1 x 1
x 1 x 1
200 200
2
2x 1 x 1 x 1 x 1
x 1
200 200 200
2
2x 1 x 1 (x 1) x 1 x 1 x 1
x 1
200
2x 1
2lim x 1 x 1 0
x 1
3.122.
x 0lim f x 0
xf x xf xf x f x f xx x4x 4x 4x4x 4x
4 4 44x 4x 4x
x 0
xf xxlim 0
4x
3.123. x 0
2 2 2 2 2 22 2 2g x x x x x x x
x x x
x 0limg x 0
x 0
g x x 2 x 2 x
x x xx x x x x x
g xx xx x
x x x
x 0
g xlim 0
x
x ,0 0,2 2
2 2 2 2 2g x g xx 2 x 2 x x x
3x 3x x 3x x 3x 3x 3x 3x
2
x 0 x 0
x
x xlim lim x 03x3x
33x
x 0
g xlim 0
3x.
3.124. f x 4x 3 g x f x g x 3 4x
x 1 x 1limf x lim g x 3 4x 0
3.125.
x 22
2 x 3
f x 5xg x f x g x x 5x 6 5x
x 5x 6
2
x 2 x 2limf x lim g x x 5x 6 5x 10
179
3.126.
2 2x 0xf x x g x x 2x x 2x
g x f x g x 1 2 xx 2x x 2x
.
x 0 x 0
2xlim f x lim g x 1 2 x 6
2x
3.127.
x 1f x xg x f x x 1 g x x
x 1.
2
x 1 x 1 x 1
x 1 x 1 g x x 2x 1 f x 2 x 1 x 1 g x x x 2lim lim lim
x 1 x 1 x 1
x 1
x 1 x 1lim
g x x 2 x 1
x 1 9
3.128.
2x 2
2
2 x 3
f x xg x f x g x x 2 x 3 x
x 5x 6.
2
x 2x 2 x 2
f x 2x g x x 2 x 3 x 2xlim lim lim g x x 3 x 3
x 2 x 2
3.129.
x 2f x 2xg x f x g x x 2 2x
x 2,
x 2 x 2
limf x lim g x x 2 2x 4 .
2
x 2
f x 2f x 8lim
f x 12 2x
x 2
f x 4 f x 2 f x 12 2xlim
f x 12 2x f x 12 2x
2x 2 x 2
f x 4 f x 2 f x 12 2x f x 4 f x 2 f x 12 2xlim lim
f x 12 4xf x 12 2x f x 12 2x
2x 2
g x x 2 2x 4 g x x 2 2x 2 g x x 2 2x 12 2xlim
g x x 2 2x 12 4x
x 2
x 2lim
g x 2 g x x 2 2x 2 g x x 2 2x 12 2x
x 2
144
13g x 2 2x 3
3.130.
x 4f x 4
2g xx 2
x 4limg x
x 4 x 4
g x x 2 f x 4 f x g x x 2 42 2
1
x 4 x 4
x 4lim f x lim g x x 2 4 0 0 4 4
2.
180
x 4
f x 4lim
x 4
0
0 1
x 4 x 4
x 4g x x 2 4 4f x 4 2lim lim
x 4 x 4
x 4
x 4
x 2 2lim g xx 4 x 4
x 4
x 4lim g x
x 4
x 4
12 1x 42 4 2 4x 2
2
.
3.131.
x f x xg x
x, x 0
x 0lim g x 1.
:
x
x f x g x x x f x g x 1x
.
:
x 0 x 0
xlim f x lim g(x) 1 1 1 1 0
x.
:
22
x 0 x 0 x 0lim f x lim f x lim f x 0 .
3.132.
2
f xh x
x 3x 2, x 1 x 2
x 2lim h x 3 2
f x h x x 3x 2 .
2g x x 4 x ,
x 2lim x 1 x 2
2
xg x
x 4
2
2x 2 x 2 x 2
h x x 1 x 2xlim f x g x lim h x x 3x 2 lim
x 4
x
x 2
3
4x 2
3.133. h x 3f x g x x f x 2g x ,
:
x 2h x
f x7
3 x h x
g x7
,
:
x 2 x 2
x 2h x 2 26lim f x lim 4
7 7,
x 2 x 2
3 x h x 6 13lim g x lim 1
7 7.
3.134. 3xg x xf x h x 1 2x g x f x t x 2 .
1 , 2 2D x x 3 ,
gD h x xt x , 2
fD 3xt x x h x .
x 0
g
2
D h x xt xg x
D x x 3
2
f
2
3xt x x h xDf x
D x x 3.
2
2x 1 x 1
3xt x x h xlimf x lim 2
x x 3
2x 1 x 1
h x xt xlimg x lim 1
x x 3
181
3.135.
x 1f x 2h x f x h x x 1 2
x 1,
x 1 x 1
limf x lim h x x 1 2 2 ,
x 2g x 3t x g x t x x 1 3
x 1
x 1 x 1
limg x lim t x x 1 3 3
x 1 x 1 x 1 x 1
x 1f x 2x h x x 1 2 2xlim x lim lim lim
g x 3x t x x 1 3 3x
h x 2
x 1
1
t x 3
3.136.
3 3x xf x x 1 g x f x g x x 1
2 2.
3
x 2 x 2
xlim f x lim g x x 1 1
2.
2 x u
x 0 x 0 u 2 u 2limf x limf 2 x limf u 1
3.137.
x 1f x x 1 1g x f x g x x 1 x 1 1
x 1
x 1 x 1limf x lim g x x 1 x 1 1 1.
3 x u
u 1x 2 x 2 u 1 u 1
f x 1 f 3 x 1 f u 1 g u u 1 u 1 1 1lim lim lim lim
x 2 x 2 u 1 u 1
u 1
u 1lim g u 1
u 1
3.138. g x f x 4x 2
x 2lim g x 2 .
f x g x 4x 2
x 2 x 2lim f x lim g x 4x 2 12 .
x 2 f x 2 f x 2 x 2 2 2x 2 f x 2x 2 f x 2
x 2 u
x 0 x 0 u 2 u 2limf x lim 2x 2 f x 2 lim 2 u 2 2 f u 14
3.139. 2g x f x x x
x 2limg x 2
2f x g x x x
2
x 2 x 2limf x lim g x x x 4 .
f x f 2 x
x 2 x 2 x 2limf x limf 2 x limf 2 x 4 .
2 x u x 2 u 0
u 0limf u 4
x 0lim f x 4 .
3.140. f f x 1 f 1 x
1 x u
x 0 x 0 u 1 u 1lim f x 1 f 1 x lim 2f 1 x lim 2f u 6
3.141.
x 1f x 2xg x f x g x x 1 2x
x 1
x 1 x 1
lim f x lim g x x 1 2x 2
x 3 u
x 2 x 2 u 1 u 1limf x limf x 3 lim f u 2
182
3.142.
2 2 2
2 2 2x 0 x 0
f 2x 3x f 2x 3x 2x 3xlim lim
2x 3x 2x 3x 2x 3x
2
2x 0
f 2x 3x xlim
2x 3x
2x 3
x
22x 3
3.143.
x 0 x 0
f 27x f 27x f 9x f 3xlim lim 64
f x f 9x f 3x f x
3.144.
22x 0
2 2 2f x f xx
x 3x f x xf x 3x x x
.
x 0
f xlim k
x,
22
x 0
f xxlim 3
x x
2 2
x 0
f xlim 1 3 k k k k 2 0
x
k 2 k 1 x 0 f x 0 .
3.145. Q x x 0
x 0 x 0
Q x 1 1 Q x 1 1 Q xlim lim
x Q x x.
x 0
Q(x) 1 1lim
Q(x) Q x 1 y
Q x y 1 x 0 y 1,
1 2x 0 y 1 y 1
Q x 1 1 y 1 y 1 1lim lim lim
Q x y 1 y 1 y y 1,
1
1 21 1
1 1 1x 0 x 0 x 0
Q x x x xlim lim lim x x
x x.
x
1
1x 0
x 1 1 1lim .
3.146. ) x 0 4f 0 0 f 1 1 0 2f 2 2f 2
x 1 4f 1 1 f 1 1 1 2f 2 f 1 1 f 1 1 2f 2 f 2 0
f 0 2 0 0 f 0 f 2 0 4f x x f 1 1 x , x .
x 2 4f 2 2 f 1 1 2 0 16 2f 1 2 f 1 7
4 4f x x 7 1 x x 8x , x .
183
x 0 x 0
f x 4 2 f x 4 2f x 4 2lim lim
x x f x 4 2
22
x 0
f x 4 2lim
x f x 4 2
x 0
f x 4 4lim
x f x 4 2
x 0
f x 1lim
x f x 4 2.
4
x 0 x 0limf x lim x 8x 0
3
3
x 0 x 0 x 0
x x 8f xlim lim lim x 8 8
x x,
x 0
f x 4 2 1 1lim 8 8 2
x 40 4 2.
3.147. x 1 u 3f x 4x 3x , 3
f x 4 x 3 x
3g x 4f x 3f x
1 2
x ,x 1 2
x x 1 2
f x f x .
x 0limf x 0 ,
2
x 0 x 0
f x x xlim lim 4 x 3 3
x x x,
x 0
f xlim 3
x
1 x 0 x 0x 0 x 0
f f x f f xg x g x f xlim lim lim lim 9
x x f x xf f x
f x u
x 0 u 0
f f x f ulim lim 3
f x u
x 0 x 0
g g x g g x g xlim lim 81
x g x x
3
x 0 x 0 x 0limg x lim f f x lim 4f x 3f x 0 ,
x 0 y 0
g g x g ylim lim 9
g x y
3.148. 1 2
x ,x 1 2
x x 1 2f x f x
1 2e f x e f x 1
xg x e x , x .
1 2
x ,x 1 2
x x 1 2x xe e 1 2x x
1 2 1 2e x e x g x g x ,
g . 1
g
1 2 1 2g f x g f x f x f x f .
f 1 1
f x y
f x y 1e f x x e y f y 1 x
f x e x , x .
184
1f fC ,C y x
fC
1f
C y x 1f
C y x
1 xf x x e x x x
e 0
f x y 1x f y .
x 1limf x 0 y 0 .
2 2 2
1 y y yf xx 1 y 0 y 0 y 0
y 1f x 1 y 1 y 1lim lim lim lim
f y 1 e e y 1 ex 1 e
y 1
y 11
3.149. 1 2
x ,x 1 2
x x 1 2
f x f x 3 3
1 2f x f x
3 3
1 1 2 2 1 2 1 2f x f x f x f x x 2 x 2 x x
f x y 1 3f x x x 2
1 3
2 2x 1 x 1 x 1
x 1f x x x 2lim lim lim
x x 2 x x 2
2x x 2
x 1
4
3x 2
3 2
2
x 2f x f x x 2 f x f x 2 x 2 f x
f x 2
2 2
x 2 x 2x 2f x
2f x 2 f x 2
x 2 x 2f x
2 2
x 2 x 2
x 2 x 2lim 0 lim
2 2
x 2lim f x 0
3.150. : 2g g x g x g x g x g x
g g g x g g g x g g x g g x
g x g x g x g x
g g g x g x g x g x g x
2
x 0 x 0
g x x x x xlim lim x
x x x x,
x 0 x 0
g x g x g x g xf xlim lim
x x
x 0
g xlim g x g x g x
x
3
0 0 0 ,
2
x 0 x 0lim g x lim x x 0 .
3.151.
x 0f x 2f x
g x f x 2f x xg xx
1
x u :
u 0 x 0
f u 2f u 2f x f xlim 3 lim 3
u x
185
x 02f x f x
h x 2f x f x xh xx
2
1 , 2
2xh x xg x
f x3
x 0lim f x 0
i.
x 0 x 0
f x 2h x g xlim lim 1
x 3
ii.
x 0 x 0
f f x f f x f xlim lim 1
x f x x
x 0 u 0
f f x f ulim lim 1
f x u
iii.
x 0 x 0
f x f x xlim lim 1
x x x,
x 0 u 0
f x f ulim lim 1
x u
3.152. x 1 u x u 1 3 2 3
f u 4 u 1 12 u 1 9 u 1 1 4u 3u ,
3f x 4x 3x , x .
3
3 3g x f f x 4 4 x 3 x 3 4 x 3 x .
x 0 x 0 x 0
f f x f f xg x f x xlim lim lim 3 3 9
x x f x x x
x u
x 0 u 0 u 0 u 0
uf x f ulim lim lim
x u
24u 3
u 3 ,
f x u
x 0 u 0 u 0
f f x f ulim lim 3
f x u
3.153. 2 2 2x 2x f x x f x 2 x xf x
2 2 2 2x 2x f x 2xf x 2 x 2 2 2 2
x 2x f x 2xf x 2 x
2 2 2 2 2 2x x f x 2xf x x 2 x x
22 2 2 2x x f x x 2 x x .
2 2 2 2
x 0 x 0lim x x lim 2 x x 0
2
x 0lim f x x 0
2 2
f x x f x x f x x
2 2
f x x x f x f x x x
x 0lim f x 0
x k u
x k u 0 u 0lim f x k limf u 0
3.154. 33 2
f x x 3x 3x 2 x 1 1.
1 2
x ,x 1 2
x x
3 3 3 3
1 2 1 2 1 2x 1 x 1 x 1 x 1 x 1 1 x 1 1
1 2f x f x ,
f 1 1
3 3
f x y x 1 1 y x 1 y 1.
y 1 3 3x 1 y 1 x y 1 1 y 1
3 3x 1 y 1 x y 1 1
3
1
3
x 1 1 , x 1f x
x 1 1 , x 1
186
23 2
x 0 x 0 x 0
x x xx 3 x 3f x 2 x 3 x 3 x x x xlim lim lim 3
xx x
x
3 3
f x x 1 1 y x 1 y 1.
x 1 y 1 0 y 1 3 3x 1 y 1 x y 1 1,
1 3f y y 1 1, y 1 1 3f x x 1 1, x 1
1 3 3
1x 0 x 0 x 0
f x f 0 x 1 1 2 x 1 1lim lim lim
x xf f x
x 0
xlim
1 1
x
23 3
1
3x 1 x 1 1
3.155. OA OB R
AKAK R
2 R 2
AB 2AK 2R
2.
OKOK R
2 R 2.
2 2 RS R
S.
0 0 0
S R 1 1 1lim lim lim 1
1AB 2 22R 1
2 2 2
2
2
.
0 0
S Rlim lim 0
AB R
0 0 0
2RAB 2 2lim lim limR R 1 R
2
.
1 1AMB AB MK 2R R OK R R R
2 2 2 2 2
2R 1
2 2
21 1OAB OA OB R
2 2
2
2 22 2
3 3
3 1R 1 1 1
AMB 2 2 3 3 2 2lim lim 2 2 1
1OAB 3R2 3 2
3.156. 1
x:
2 2
2
2 2
1 1 x 1 1 1 x 1f x f
x 3 x 33x 3x
2
x 0
1 1lim x f
x 3
187
3 2
2x 0 x 0
1 1 3xx f 3x x f 3
10x x 3xlim lim
x 3x xx
x
3.157.
x 0
2 2 2 2f x 4 4xf x 4 x f x 4xf x 4 x 4
2 2
2 2
f x f x x4 4
xx x
2 2
2 2x 0 x 0
f x f x xlim 4 lim 4
xx x,
x 0
f xlim k
x
22k 4k 4 k 2 0 k 2
2x 0 x 0 x 0
f x f x f x x 1 1lim lim lim 2 1
x x 2 x x x 2 2x 2x
x 0 u 0
f x f ulim lim 2
x u
188
3.171.
x 3 x 3
2x 7 1lim lim 2x 7
x 3 x 3
x 3lim 2x 7 1 0 ,
x 3
1lim
x 3
x 3lim x 3 0 x 3 0 x 3 .
2 2x 1 x 1
2x 1 1lim lim 2x 1
x 1 x 1
x 1lim 2x 1 3
2x 1
1lim
x 1
2 2x 2 x 2
x 3 4 1lim lim x 3 4
x 4x 4 x 2,
x 2lim x 3 4 5
2x 2
1lim
x 2
x 1 x 1
x 3 2 1lim lim x 3 2
x 1 x 1,
x 1lim x 3 2 2
x 1
1lim
x 1
2
3x 2 x 2
x 2x 5x 6lim lim
x 2
3
x 3
x 2
22 x 2
1lim x 3
x 2,
x 2lim x 3 1
2x 2
1lim
x 2
2 2x 1 x 1
x 8 2 1lim lim x 8 2
x 2x 1 x 1,
x 1lim x 8 2 1
2x 1
1lim
x 1
3.172.
2
x 5 x 5
3x 1 3x 1 1lim lim
x 5 x 5x 25
2
x 5 x 5
3x 1 3x 1 1lim lim
x 5 x 5x 25
2
4x 5 4x 5 1f x
x 2 x 2x 4, x 2 .
x 2
4x 5 13lim 0
x 2 4,
x 2
1lim
x 2
x 2
1lim
x 2
x 2
lim f x ,
x 2
lim f x x 2
lim f x .
2
2
x 2 x 2
x x 1lim lim x x
x 2 x 2,
2
2
x 2 x 2
x x 1lim lim x x
x 2 x 2,
189
3 2x 1 x 1
x 1 2x x 1 2x 1lim lim
x 1x 1 x x 1,
3 2x 1 x 1
x 1 2x x 1 2x 1lim lim
x 1x 1 x x 1
2
x 5 x 5
x 1 x 1 1lim lim
x 2 x 5x 7x 10,
2
x 5 x 5
x 1 x 1 1lim lim
x 2 x 5x 7x 10,
2 2 2x 0 x 0 x 0
1 1 x 1 1lim lim lim x 1
x x x x
3.173.
2
2x 1 x 1
x 1x 3x 4lim lim
x 1
x 4
x 1 x 1,
x 1
x 4lim
x 1,
x 1
x 4lim
x 1
2 2
3 2 2x 1 x 1
x 6x 10 x 6x 10 1lim lim
x 1x x x 1 x 1
3 2 2x 3 x 3 x 3
4x 7 4x 7 4x 7 1lim lim lim
x 3xx 6x 9x x x 3
2
3 2x 3 x 3
x 3x 9lim lim
x 3x 9x 27
x 3
2
x 3 x 3
x 3
1lim
x 3
x 3
1lim
x 3,
x 3
1lim
x 3
3.174.
22
x 0 x 0
2x 3 1 x 1 x2x 3lim lim
1 x 1 x 1 x 1 x 1 x 1 x
2
x 0
1lim 2x 3 1 x 1 x
2x
23 3
3x 0 x 0
2x 5 1lim lim 2x 5 x 27 3 x 27 9
xx 27 3
2
2
x 1 x 1
x 4x 1lim lim x 4x
x 1 x 1
)
2 2
2
x 1 x 1 x 1
x x 3 x x 3 1lim lim lim x x 3
1 x 1 x 1 x
x 2 x 2 x 2
x 2x 2 1lim lim lim x 2
x 2 x 2 x 2
x 0 x 0 x 0
x 2 2 x 3 x 2 2x 6 1lim lim lim x 8
x 1 1 x 1 1 x
x 0
1lim x 8
x,
x 0
1lim x 8
x
190
3.175.
x 0 x 0
3x 3x 1lim lim 3
x x 3x x
x 0
3xlim 1
3x
x 0
1lim
x
x 0 x 0
x 1lim lim x
x x
x 0lim x
x 0
1lim
x
x 0 x 0
x 1lim lim x
x x
x 0
1lim
x
x x
2 2
2x 1lim lim 2x
x x
x
2
lim 2x 2
x2
1lim
x,
x x
2 2
2x 1lim lim 2x
x x,
x2
1lim
x,
x 0 x 0
2 x 1lim lim 2 x
1 x 1 x
x 0lim 2 x 2 ,
x 0
1lim
1 x
x 0 x 0
x 3 1lim lim x 3
x 1 x 1
x 0lim x 3 3 ,
x 0
1lim
x 1
x 0 x 0
2 7 x 1lim lim 2 7 x
3 x 3 x,
x 0
lim 2 7 x 5 ,
x 0
1lim
3 x
3.176.
2
2 2x 3 x 3 x 3
2x x 4 32x 3 2x 8x 3 1lim lim lim
x 3 x 4 x 3x 7x 12 x 7x 12
2
x 3
2x 8x 3lim 9
x 4
x 3
1lim
x 3,
x 3
1lim
x 3
2 2x 1 x 1
1 1 1 1lim lim
x 1 x 2 x 1 x 2x 3x 2 x x 2
2x 1
2x 1lim
x 1x 4
2x 1
2x 2lim
3x 4
x 1
1lim
x 1,
x 1
1lim
x 1
3.177.
x 4 x 4
9 9lim lim
x x 2x 4 x 8 x x 4 2 x 4
x 4
9lim
x 4 x 2
2x 4 x 4
9 x 2 1lim lim 9 x 2
x 4 x 4 x 4
x 3 x 3
2x 1 2x 1lim lim
x x 1 6 3 x 1 2x x 1 x 3 2 x 3
x 3 x 3
2x 1 x 1 22x 1lim lim
x 3 x 1 4x 3 x 1 2
2x 3
1lim 2x 1 x 1 2
x 3
191
3.178.
x 0 x 0
1lim f x lim
x,
x 0 x 0
2lim f x lim
x
3.179.
2
24x 1 1g x f x 4x 1
f x g x
2
x 2 x 2
1lim f x lim 4x 1 0
g x
3.180.
2
2
1g x x 2x 5 f x f x g x
x 2x 5
2x 2 x 2
1lim f x lim g x
x 2x 5
3.181.
g x x 8 3h x ,
x x 1
x 1limh x .
h x x x 8 3 1 xh x xg x h x x 8 3
x 1 1 xx 8 3,
x 1limg x .
3.182.
x 22
2
1x 2 f x g x f x g x
x 2
2x 2 x 2
1lim f x lim g x
x 2
2
2
2x 2 x 2
f x3f x 5f x 7
lim limf x 2
53
f x
0
2
7
f x
0
2f x
2
21
f x
03
3.183.
x 1
x 1
4f x 3xg x x
2xf x 3g x h x
24 3x
D 12 6x2x 3
,
f
x 3xD 3 x 3xh x
h x 3,
g
4 xD 4h x 2x x
2x h x.
2
3 x 3xh xf x
12 6x
2x 1 x 1
3 x 3xh xlimf x lim
12 6x
2
4h x 2x xg x
12 6x
2x 1 x 1
4h x 2x xlimg x lim
12 6x.
3.184. f f x f x 1 x .
1
x 0 x 0
lim f x lim f x 2
192
xx 0
0 .
2
0 0 0
lim f lim f lim f .
x 0
lim f x .
3.185. x 0
2
4 2
4
4x 2x 1x g x 4x 2x 1 g x
x.
2
2
4 4x 0 x 0
4x 2x 1 1lim lim 4x 2x 1
x x
x 0limg x .
x 0limg x g x 0 .
x 0
1lim 0
g x
2 2
2
2x 0 x 0
4g x 2g x 1 4g xlim 4g x 2g x 1 2g x lim
4g x 2g x 1 2g x
x 0
2
2
2g x 1lim
2 1g x 4 2g x
g x g x
x 0
2
2g x 1lim
2 1g x 4 2g x
g x g x
x 0
g x
lim
12
g x
g x
2
2 1
4 22 14 2
g x g x
.
3.186.
2
2x 3x 1f x x 3x
x 1 x 1
fD {1} .
2
x 1lim x 3x 4
x 1
1lim
x 1 .
4 0 4
x 1limf x
4 0 4
x 1limf x
4 0 4
2x 3x 4
f xx 1
x 4 x 1
x 1.
x 1 x 1 x 1
x 4 x 1lim f x lim lim x 4 5
x 1
x 1 x 1 x 1
x 4 x 1lim f x lim lim x 4 5
x 1.
x 1
limf x
193
3.187.
2 1f x x 2x
x 1, x 1.
2
x 1lim x 2x 1
x 1
1lim
x 1.
1
x 1limf x 1
x 1limf x
1
2
x 1 x 1
x 1lim f x lim
x 1 0 ,
2
x 1 x 1
x 1lim f x lim
x 1 0
x 1limf x 0
2
1f x x
x 2, x 2 .
x 2lim x 2
2x 2
1lim
x 2
2
x 2lim f x 2
x 2lim f x
2
x 2 x 2
x 2lim f x lim
2
x 2 ,
x 2 x 2
x 2lim f x lim
2
x 2
3.188. x 1
2
2
3x x 2f x
x 1
23x x 2 1
x 1 x 1.
:
x 1lim x 1 0
2
x 1
3x x 2 3 2lim
x 1 2:
3 20 3 2 0
2
x 1
1lim
x 1,
x 1
lim f x ,
x 1
1lim
x 1
x 1
lim f x x 1
limf x .
3 20 3 2 0
2:
x 1
lim f x
x 1
lim f x ,
x 1
limf x .
3 2 0 2 3
2
2
3 x 1 x 1 x 13x x 3f x
x 1 x 1x 1
x 1
3x 3
x 1 x 1
3x 3f x
x 1
x 1limf x
x 1
3x 3 6lim
x 1 2.
3.189.
xf x
x 1 x 1 2
x 1.
x 3lim x 3 ,
x 3lim x 1 2 :
2 0 2 :
x 3 x 3
x 3lim f x lim
2x 1
2 0 2
x x 1 2x 1f x x x 1 2
x 3 x 3x 1 2
x 3
lim x x 1 2 3 4
x 3
1lim
x 3,
x 3
1lim
x 3
194
3 0 3
x 3
lim f x ,
x 3
lim f x x 3
lim f x
3 0 3
x 3
lim f x ,
x 3
lim f x x 3
lim f x
3 0 3
x 3 x 1 2xf x x 1 2
x 3x 1 2
x 3 x 3lim f x lim x 1 2 4
3.190.
2 22 22 2
x x 2 2
x xxlim lim
xx x
2 2
x
xlim
2
2
x2 22 2
xlim
x x x x
.
0 x x 0
2
x
xlim
x
x2 22 2
xlim 0
x x
0 x x 0
2 2 22 2 2
x x x
x xlim lim lim
x x
2
x xlim lim 1
0
2 2 2
x x x
x xxlim lim lim 1
x x x.
3.191.
x 0lim x x ,
x 0
1lim
x,
x 0
1lim
x
x 0
lim f x .
x 0 x 0
x 1 xlim f x lim
x x
3.192.
4 4 4 4
6 6 6 6 6 2
f x g x f x f x 10
f x g x f x g x f x f x.
2x 0
1lim 0
f x
4 4
6 6x 0
f x g xlim 0
f x g x
3.193.
2 1f x x x
x 2, x 2 .
2
x 2lim x x 2 4
x 2
1lim
x 2
195
2 4 0
x 0lim f x 2 4 0
x 0lim f x ,
x 2lim f x 2 4 0 2 4
x 2 x 2
x 2lim f x lim
x 2
x 2 4
x 2 x 2
x 2lim f x lim
x 2
x 2 4
4 4 4 2 4 4 4 .
3.194.
3 2x x 2x 8 1
f xx 2 x 2
, x 2 .
3 2
x 2
x x 2x 8 8 4 4lim 2 1
x 2 4
x 2 x 2
1 1lim , lim
x 2 x 2,
2 1 0 2 1 0 x 2
lim f x .
x 2lim f x 4 2 1 0 2 1
x 2 x 2
x 2lim f x lim
2x x 4
x 2
4 6 54
4 2x 2
52 1 4
2
3.195.
2 2 2 2
2 2
x 4 x 4 x 4 x 4 1f x
x xx, x
22 2
x
2x 4 x 4lim
x 2,
x
1lim
x,
x
1lim
x
22
02
22
02
x
lim f x .
xlim f x 2 4
22
0 22 2
22 2
2 2x x x
xx 2 xlim f x lim lim
x x
0
x 2 4 0 2 1.
3.196.
2x 3 x 3
x 1 x 4 1 x 1 x 4 1lim lim
x 3 x 3x 9.
x 1 x 4 1 1f x
x 3 x 3, x 3 .
x 3
x 1 x 4 1 4 1lim
x 3 6,
x 3
1lim
x 3,
x 3
1lim
x 3,
4 1
06
4 1
06
x 3
lim f x
x 3lim f x
4 10 1 4
6
x 3 x 3
5 1 x 3lim f x lim
x 3
5 1
6x 3.
5 19 11
6 1 4 11 43
196
3.197.
3 2 1f x x x 6x 8
x 1, x 1.
3 2
x 1lim x x 6x 8 14
x 1
1lim
x 1 14 0
x 1limf x 14 0
x 1limf x .
x 1limf x 14 0 14 .
x 1 x 1
x 1lim f x lim
2x 14x 8
x 1 22 ,
x 1 x 1
x 1lim f x lim
2x 14x 8
x 1 22 22 22 22 , 0 .
22 14 36
3.198.
2x 2 x 2 x 2
x 3 x 3 5 x 3 x 3 5 1lim f x lim lim
x 1 x 2x 3x 2, x 1,2
x 2
x 3 x 3 5lim 5 5
x 1,
x 2 x 2
1 1lim , lim
x 2 x 2,
5 5 0 5 5 0 x 2
lim f x
x 2lim f x 7 ,
5 5 0 5 5
x 2 x 2
x 2lim f x lim
6 5
x 1 x 2 6 5 .
6 5 7 2 5 5 2 5
3.199.
2 1f x x x 1
x 3, x 3 .
2
x 3lim x x 1 8 3 ,
x 3
1lim
x 3,
x 3
1lim
x 3,
8 3 0 x 3
lim f x .
8 3 0 3 8
x 3 x 3
x 3lim f x lim
x 3
x 3 6 5 1,
3 8 5
3.200.
x 1 x 1
lim f x lim f x . x 1:
2
2
x 1f x f x x 1 x 1
x 1
2
x 1 x 1
lim f x x 1 lim x 1 0 1 1,
x 1 x 1
x 1lim f x lim
x 1
1
2x 1
x 1 x 1
lim f x lim ln x ln 1
1
21 1
ln 1 1 e 12 e
197
3.201.
2x 0 x 0 x 0
x x x xlim f x lim lim
x x 1x x x
x 1
2x 0 x 0 x 0
x x x xlim f x lim lim
x x 1x x x
x 1
0 2 1
3.202.
2
2
f xx 1: g x f x g x x 1
x 1
2
x 1 x 1limf x lim g x x 1 .
2
x 1 x 1limf x lim x 2x 1 1.
2 22
2 2 2x 1 x 1 x 1 x 1
x 1x 1f x x 2x 1lim lim lim lim
x 1 x 1 x 1
21 2
2
x x ... x 1
x 1
2
3.203.
x ,0 0,2 2
2
2xf x x f x 1 x
1 x
2 2
x 0 x 0lim x lim f x 1 x 0 .
2 2 2 22 2
x 0 x 0 x 0 2 2
x x 1 xx
lim f x lim lim1 x 1 x 1 x x
2 2 2
2x 0 x 02 2 22 2
2
x 1 x 1 x 1lim lim
x1 x x xx
,
11 1 1
3.204. x 1
x xf x x x f x x 1
x 1
x 1 x 1lim x x lim f x x 1 1 0 1 .
1
x 1 x 1 x 1
x x 1 x 1x 1 xlimf x lim lim
x 1 x 1
x 1
x 1lim x
2 3x x ... 1
x 1
1 4 5 4
3.205. x 1
2
2 3
3
x g x 3f x x g x 3 f x x 1
x 1
2 3
x 1 x 1lim x g x 3 lim f x x 1 1 .
198
2 2
3 3x 1 x 1
2x x 3 2x x x 3lim lim
x 1 x 1
x 1
x 1lim
2x 3 x 1
x 1
2
51
3x x 1 2 1.
3.206.
4 2
2
x x xf x
x x 1 1, x 0
x 0limf x .
4 2
2 2
xx xf x
x x 1 1 x x 1 1
2 2 2
2 2
x x 1 x x 1 1 x x x 1 1
x x 1 1 x x 1 1
2 22x x 1 x x 1 1 x x x 1 1
x x 1 x x 1 x.
2 22
x 0 x 0
x x 1 x x 1 1x x x 1 1
lim f x lim 2 2x x 1 x x x 1
.
x 0
lim f x .... 2 2 2 2 2 2 1.
3.207. x 2
2
2x xf x f x x 2 x x
x 2
2
x 2 x 2
lim f x x 2 lim x x 0 6 6 ,
2
x 2 x 2 x 2
x 2x x 6lim f x lim lim
x 2
x 3
x 2 5
x 2
2
24x kxf x f x x 2 4x kx
x 2
2
x 2 x 2
lim f x x 2 lim 4x kx 0 16 2k 16 2k ,
2
x 2 x 2 x 2
x 24x kx 16 2klim f x lim lim
x 2
4x 8 k
x 2 16 k .
5 16 k k k 11 6 .
3.208.
2
2
g xf x x 2 g x f x
x 2
2x 1 x 1
g xlimf x lim
x 2
1
uf x
x 1 u 0 u 0
1 1 ulim f x lim u lim 1
f x u u
x 1limf x g x f x
x 1limg x
199
3.209. y x 3 38f 2x f 2x f 2x 48x f 2x 8x .
2x u 3f u u 3
f x x , x .
i.
2 6x 0 x 0
1 1lim lim
f x x
ii.
3
9 63x 0 x 0 x 0
f x x 1lim lim lim
x xf x
1 2
x ,x 1 2
x x 3 3
1 2 1 2x x f x f x f 1-1.
1 3f x 1 u x 1 f u x u 1 x 0 u 1
1 1
3x 0 u 1 u 1
f x 1 f x 1 2 u u 2 u 1lim lim lim
x u 1 u 1
32
u 1
u 1u u 1
2u 1
1 u 1lim
u u 1 u 1
2
1 1 1
3 6 2u u 1 u 1
3.210. 5 3 2 4 2 23f x f x f x x 9 f x 3f x f x 1 x 9
2
4 2
x 9f x
3f x f x 1
4 2
4 2
13f x f x 1 1 1
3f x f x 1,
2
2 2 2
4 2
x 9f x x 9 x 9 f x x 9
3f x f x 1
x 3lim f x 0 .
x 6 x 6 x 6f x f x f x f x f x f x
x 3 x 3 x 3
x 0
x 6lim f x 0
x 3.
2x 0
6lim
f x
x 0limh x
3.211.
2 xx 22 x e
x 2 f x e f xx 2
2 x
x 2
elim
x 2
x 2lim f x
i.
3 2 3
3x 2 x 2
7 3lim 5f x 7f x 3 lim f x 5
f x f x
ii.
f x f x f x f x f x f xe f x e f x e e e f x e
f x uf x u
x 2 u ulim e lim e 0
f x
x 2lim e f x 0
200
3.241.
3 2 3
xxlim 2x 3x 7x 100 lim 2x
2 2
x xlim 4x 5 lim 4x
4 2 4
x xlim 2x 4x 3x 6 lim 2x
3.242.
2 2
3 2 3x x x
4x 7x 8 4x 4lim lim lim 0
5x5x 6x 9x 5 5x.
3 2 3
3 3x x
18x 6x 9 18xlim lim 3
6x 8x 12 6x.
5 5
3
2 2x x x
4x 7x 8 4xlim lim lim 2x
2x 8x 4 2x.
3.243.
2 2 3 2 3
2 2x xx
3x x 2 x x 2 x 3x 2x 4 xlim lim lim
x 1x 1 x 1 2x
4 2 5 4 3 2 5
3 4 3x x x
x 5x 7 x 3 2x x 3x 6x 2x 4 2xlim lim lim
x 1x 1 x x x 1 4x
3.244.
3 3 2 3
2 2x x x
x 7x 2 3x 3x 3x 1 3xlim 4x 3 lim lim
x 1 x 1 2x
4 2 5 4 3 2 5
3 4 3x x x
x 5x 7 x 3 2x x 3x 6x 2x 4 2xlim lim lim
x 1x 1 x x x 1 4x
3.245.
2 2 2 2
x xlim 4 x 2x 5x 3 lim x 4 2x 5x 3
2 2
x xlim x 5x 1 lim x
x x xx
3 x 2 x 1 3x 6 x 1 2x 5 2 xlim lim lim lim
x 1 2 x 1 2 x 1 x 2
2 2 2 2
2 2 2 2x x x x
1 x 3 x 1 x 1 3x 3 x 3x 2 xlim lim lim lim 1
x x x x x x x
2 2 2 2
2 22x x x x
x x 3x 5 x x 3x 5 x 2x 5 xlim lim lim lim
2 x x 2 x x 42 x x 2 2x
1
3.246.
2 2
2x x x
f x 3x 4x 1 3 xlim lim lim
x x 2x 2x
3 ,
x x x
10x 1 10 xlim f x 3x lim lim
x 2 x 10
201
3.247. 1
2 2
x xlim 1 x 3x 4 lim 1 x ,
1
2 2
x xlim 1 x 3x 4 lim 1 x
1
2
x x xlim 1 x 3x 4 lim 3x 4 lim 3x
, 3 3,
2 3 2 2 3
x xlim 9 x 4x x 1 lim 9 x
3,3 ,
2 3 2 2 3
x xlim 9 x 4x x 1 lim 9 x .
3
2 3 2 2 2
x x xlim 9 x 4x x 1 lim 4x 3x 1 lim 4x
3
2 3 2 2
x xlim 9 x 4x x 1 lim 4x 3x 1
2
xlim 4x
3.248. 1
2
x x
1 x x 1 x 1lim lim 1
2 x 5 x 5,
2
2 2
x x
1 x x 1 x 2x 1lim lim
2 x 5 5,
,1 2,
2
x x
1 x x 1 1lim lim x
2 x 5 2
1,2
2
x x
1 x x 1 1lim lim x
2 x 5 2
1
2
2 2x x x
1 x 3x 3x 1 3 xlim lim lim
1 x 2x 3 2x 2x 3 22x
0 ,
1
2 2
x x
2x 3x 1 2xlim lim
2x 3 2 x ,
1
2 2
2x x
1 x 3x 1 xlim lim
1 x 2x 3 21 x
1
1
2
2 3 2 2 2
2 2x x x
4 x 3x x 1 3x 2x 1 3 xlim lim lim
2 x 2x 1 4x 2x 1 24 x
3
4,
2
2 3 2 2 2
2x x x
4 x 3x x 1 3x 2x 1 3xlim lim lim
2x 12 x 2x 1 2 x ,
2
2 3 2
2x x
2 24 x 3x x 1lim lim
2 x 2x 1
3x
22
x
x
, 2, 2lim 2 x
, 2
202
0
4 2 2 2
32 4 3x x x
x 3x 7x 5 3x 7x 5 3 xlim lim lim
3x 14x 3x 14 33x
0 ,
1
4 2 4 2 4
32 4 3x x x
x 3x 7x 5 x 3x 7x 5 xlim lim lim
3x 14x 3x 14 33 x
,
0, 1
4 2
2 4 3x x
x 3x 7x 5lim lim
x 3x 14
4x
41 x
1
1
3.249. 1
2 3 2 2 2
2 2x x x
1 x 1 x 1 2x 1 2xlim lim lim
3x 2x 2 x 2 3 x ,
1
2 3 2
2 22 2x x x
1 x 1 x 1 1 1lim lim lim 0
2x x 2 2xx 2 x 2,
0
2 3 2 3 2 3
2 2x x x
1 x 1 x 1 x x 1 xlim lim lim
2x 2x 2 x 2
2
2 x ,
1, 0
2 3 2
2 2x x
1 11 x 1 x 1lim lim
x 2 x 2
3x
12
x
x
1lim x
, 1,0
, , 1 0,
3.250.
2 2 1 2
1
3 3 3x x x x
6x 7x 1 6x 7x x 6xlim x lim lim lim 3x
2x 6x 2 2x 6x 2 2x.
1 0 1
2 ,
1
xlim 3x 2 1,
1
xlim 3x .
1
1
xlim 3x 3 0
1
x x
1lim 3x lim 3 0
x
3.251.
2 3 2 2 2 2
2
2 x 5 5 6 x 7 x 2 2 5f x
x 3x 2, x 1,2
, 1 0,
2 3
x x
2 xlim f x lim
2x
,
1,0
2 3
x x
2 xlim f x lim
2x
,
1 0
2 2
2x x x
6x 7x 5 6 xlim f x lim lim
x 3x 2 2x
6
3.252. 4 2
3
x x
4 xlim f x lim
22 x
203
4 2
22
2x x x
2
2 18 48 4 x 2x 1 x xlim f x lim lim 4
6 32 x 6x 32
x x
4 2
x x x
2x 1 2x 1lim f x lim lim
6x 3 6x 3
3.253. 1
3
x x
1 xlim f x lim
2x
xlim f x 4 1.
2 2
2x x x
2
4
x 4x x xlim f x lim lim5 1x 5x 1
1x x
3
3.254.
21 x 1 x 2
f xx 1
, x 1.
1
2
x x
1 xlim f x lim
x
xlim f x 4 1,
x x x
2
x 2 xlim f x lim lim1x 1
1x
4
3.255.
22 x 2 2 x 7x 3f x x 2
x 2 x 2
0
2
x x
xlim f x lim
x
0
2
x x
xlim f x lim
2
0
x x
2x 7lim f x lim
2
0
x x x
72 22 2 x 7 2 2xlim f x lim lim
2x 2
x
2 24 1
3.256.
3 2
2 2
x 2 x 2x 2 xf x 2x
x 1 x 1, x 1.
0
3
x x
2 xlim f x lim
2x
,
0
2 2
2x x x
2x 2 xlim f x lim lim
x 1 2x
2
204
3.257.
22 4 1 x 2 2 x 34x x 3f x x
4x 1 4x 1
1
x x
4lim f x lim
21 x
4 x
1
x x x
32 1 22 1 2 x 3 2 4xlim f x lim lim
14x 1 44
x
2 4 73
4 2
3.258.
4 3 23
2
2 2
x 1 x x x 2x 2f x x x
x 1 x 1
0
4
x x
xlim f x lim
2
2x
xlim f x 2 0 .
3 2
2x x
1 x x x 2lim f x lim
x 1 1
xlim f x ,
1
2 2
2x x x
2
2
x 2 xlim f x lim lim1x 1
1x
2 .
3.259.
2 2
2 2x x x
g x x x 1 x 1lim lim lim
x 22x 3x 2x
3 2 3
3x x x
f x x x 2 xlim lim lim
x 2x x 32 x
1
2,
1
2.
3 2 2 2
2 2x x x x
2
1 42
x x 2 1 2 x x 4 x xlim f x x lim x lim lim22 22x 1 4x 2
4x
1 2
2.
2
x x x x
22 32 3 x 2x x 1 1 2 3xlim g x x lim x lim lim
62x 3 2 4x 6 44
x
2 3 71
4 2.
3.260.
2 2
2x x
6 2lim 9x 6x 2 lim x 9
x x
2x
6 2lim x 9
x x
2x
6 2lim x 9
x x
xlim x
2x
6 2lim 9 3
x x.
205
2x x
x3x 5
lim limx x 1
53
x
x
2
31 1
1x x
2 2
x x x
1 xx 1 x 2x 1 x 2 x
lim lim limx 1 x 1
2
1 21 1
xx
x
21
1x
54 4 3
32 2x x
x16x 3 x 2
lim limx 1 x 4
44
316 x
x5
2 5
1 2
x x
x 2
11 x
x
33
21 4
x x
3.261.
2
x x
xx 4 x 1
lim limx 3
2
4 11 1
xx
x
03
1x
2
2x x
xx x 3x 4
lim limx x 3x 4
2
3 41 1
x x
x
2
03 4
1 1x x
3
3 2
2x x
x2x 3x 5x 1
lim limx 2x 5
2
2 3
3 5 12
x x x
x
2
2 51
x x
2
2x x
xx 3x 7 5x
lim limx 6x 8 x
2
3 71 5
x x
x
2
36 8
1 1x x
3.262.
2
2x
2 1lim x 9 3x
x x
2x
2 1lim x 9 3x
x x
2x
2 1lim x 9 3
x x
2
xlim 9x 2x 1 3x
2 2
2x
9x 2x 1 3x 9x 2x 1 3xlim
9x 2x 1 3x
2 2
x x
2 2
9x 2x 1 9x 2x 1lim lim
2 1 2 1x 9 3x x 9 3x
x xx x
206
x
x
lim
12
x
x
2
2 19 3
x x
x
2
12
2 1xlim6 32 1
9 3x x
2
2x x
1 2lim x x 2 x lim x 1 1
x x
2
x x
3lim x 3x 2x lim x 1 2
x
2
2
x x
4xlim 4x 3x 5 2x lim
23x 5 4x
2 x
x
lim4x 3x 5 2x
53
x
x
2
3
43 54 2
x x
2 2 2 2
2 2
2 2x x
x 2 x 1 x 2 x 1lim x 2 x 1 lim
x 2 x 1
2
x
xlim
22 x
x
2 2 2 2
1 3lim 0
2 1 2 1x 1 x 1 x 1 1
x x x x
2
2
x x
xlim x 1 x lim
21 x
2
2
x
1lim 0
1x 1 xx 1 1
x
2 2 4 4
2 2
2 2x x x
x x 2x 2 x xlim x x 2x 2 x lim lim
x x 2x 2 x
3 2 42x 2x x
2 2
2
2 2x 1 x
x x
3
x
x
lim
2
22
x
x
2
2 21 1
x x
3.263.
32 33 3
2 3 2 3x x x
5 2 5 2lim x 5 8x 2 lim x 1 x 8 lim x 1 8
x x x x
3.264.
2 2 2
xlim x x 2 4x 3x 1 x 5x 10
2 2 2
xlim x x 2 4x 3x 1 x 5x 10 2x x x
2 2 2
xlim x x 2 x x 5x 10 x 4x 3x 1 2x
2 2 2 2 2 2
2 2 2x
x x 2 x x 5x 10 x 4x 3x 1 4xlim
x x 2 x x 5x 10 x 4x 3x 1 2x
207
x
2 2 2
x 2 5x 10 3x 1lim
1 2 5 10 3 1x 1 x x 1 x x 4 2x
x x xx x x
x
x
lim
21
x
x
2
x
1 21 1
x x
105
x
x
2
x
5 101 1
x x
13
x
x
2
3 14 2
x x
1 5 3 9
2 2 4 4.
3.265.
2 2 2 2
x xlim x x 1 9x 2x 4x 1 lim x x 1 x 9x 2x 3x 1
2
x
xlim
2x 1 x
2
2
9x
x x 1 x
22x 9x
21
9x 2x 3x
x
x
lim
11
x
x
2
2 x
1 11 1
x xx
12
9 3x
1 2 11
12 6 6
2 2 2
xlim 4x 2 9x 1 25x x 1
2 2 2
xlim 4x 2 2x 9x 1 3x 25x x 1 5x
2
x
4xlim
22 4x
2
2
9x
4x 2 2x
21 9x
2
2
25x
9x 1 3x
2x 1 25x
225x x 1 5x
x
2 2
x2 1
lim2 1
x 4 2 x 9 3x x
11
x
x
2
1 10 0
10 10125 1 5
x
3.266.
2 2 2 2
x xlim x x 5 9x 1 4x 3 lim x x 5 x 9x 1 3x 3
2
x
xlim
2x 5 x
2
2
9x
x x 5 x
21 9x
23
9x 1 3x
x
x
lim
51
x
x
2 2
13
1 5 11 1 x 9 3
x x x
208
1 7
0 32 2
2 2 2 2
x xlim 4x x 1 x 3 x 5 lim 4x x 1 2x x 3 x 5
2
x
4xlim
2x 1 4x
2
2
x
4x x 1 2x
23 x
25
x 3 x
x
x
lim
11
x
x
2 2
35
1 1 34 2 x 1 1
x x x
1 19
0 54 4
3.267.
2 4 2
2 4
x x x 2 4
x x 3 2xlim g x lim x x x 3 x 2 lim x
x x 3 x 2
4 2 4
x x2 4
x 3 x xlim x lim x
x x 3 x 2
43 x
2 2 4 2
4
3x x 1 x 2 x 3 x
x
x
3 xlim
x
2
4 4
0
3 31 1 2 x 1 1
x x
2 2
4 4x x x
3 3lim g x lim x x x 1 x 2 lim x x 1 1 1 x 2
x x
2
4x
3lim x 1 1 1 2
x
3.268.
3 552 23 2 2 3
3x x x
32 3
1 1 1 xx x 1 xx x 1 x x x x
lim lim lim2x x 1 7 1 1
2x x 7x x
3 52 2 3
1 1 11
x x x
x
32 3
1
21 1 72
xx x
3.269.
2
2
f xh x f x h x 9x 6x 3 3x
9x 6x 3 3x,
xlim h x 2 ,
x 0,
2
2
g xt x g x t x x 2x 3 x
x 2x 3 x,
xlim t x 1.
2
x x 2
h x 9x 6x 3 3xf xlim lim
g x t x x 2x 3 x
209
2
x
h x 9xlim
t x
26x 3 9x
2
22
x 2x 3 x
x9x 6x 3 3x 22x 3 x
x
xh x
limt x
36
x
x
2
x
6 39 3
x x
2
2 31 1
x x
x
2 1 1 23
2x
3.270.
2
2x x
1lim x 1 x lim x 1
x
2x
1lim 1 1
x
1
2
xlim x 1 x 1
2
xlim x 1 x .
1
2
2
x x
xlim x 1 x lim
21 x
2 x
2
1lim 0
1x 1 xx 1 1
x
2
2x x
3 2lim 4x 3x 2 x lim x 4
x x.
2x
3 2lim 4 2
x x
2
2
xlim 4x 3x 2 x ,
2
2
xlim 4x 3x 2 x .
2
2
2
x x
4xlim 4x 3x 2 2x lim
23x 2 4x
2
4x 3x 2 2x
x
x
lim
23
x
x
2
3
43 24 2
x x
2
x x
2lim x 2x 2 x lim x 1 2
x.
x
2lim 1 2 1
x:
1
2
xlim x 2x 2 x
1
2
xlim x 2x 2 x .
1
2
2
x x
xlim x 2x x lim
22x x
2 x
2 xlim
x 2x xx
12
1 1x
210
2
2x x
3 2lim 4x 3x 2 x lim x 4
x x.
2x
3 2lim 4 2
x x : 2
2
xlim 4x 3x 2 x ,
2
2
xlim 4x 3x 2 x 2
2
2
x x
4xlim 4x 3x 2 2x lim
23x 2 4x
2
4x 3x 2 2x x
x
lim
23
x
x
2
3
43 24 2
x x
3.271.
2
2x x x
3lim f x lim x 3 x lim x 1
x.
2x
3lim 1 1
x, : 1
xlim f x ,
1
xlim f x
xlim f x 1.
2
2
x x x
xlim f x lim x 3 x lim
23 x
2 x
2
3lim 0
3x 3 xx 1 1
x
3.272.
2
x x x
xf x 4 x x
lim lim limx x
2
41
x
x 1 1 0
3.273.
2
2x x
1 5lim x x 5 x lim x 1
x xx.
2x
1 5lim 1 1
x xx 1,
2
xlim x x 5 x 1
2
xlim x x 5 x ,
2
xlim x x 5 x 4 1 :
2
2
x x
xlim x x 5 x lim
2x 5 x
2 x
x
limx x 5 x
51
x
x
2
1
21 51 1
x x
1 9
42 2
211
3.274. 2f x x 5x 10 x , x 0, .
2x x
5 10lim f x lim x 1
x xx.
2x
5 10lim 1 1
x xx
1
xlim f x
1
xlim f x
xlim f x 3 1.
2
2
x x x
xlim f x lim x 5x 10 x lim
25x 10 x
2x 5x 10 x
x
x
lim
105
x
x
2
5 13
2 25 101 1
x x
3.275.
2 2
2x x
1 1lim 1 x x x x lim x 1 1
x xx.
2x
1 1lim 1 1 2
x xx
2
2 2
xlim 1 x x x x
2
2 2
xlim 1 x x x x .
2 2
xlim 1 x x x x 0 2 .
2 2 2 2
x xlim 1 x x x 2x lim 1 x x x x x
2
x
1 xlim
2x
2
2
x x
1 x x
2x
2x x x
x
2
1 xlim
1x 1 1
xx
1
211 1
x
1 1
02 2
.
212
3.276. 1 f x x fA 0, 1.
2 1
0 1 4 1 0 3 2 2 1 02
3
2
1 0 1 3
2.
2
2x x x
1 1lim f x lim 1 x x 1 lim x 1
x x
2x
1 1lim 1 1 0
x x.
3.277.
2
2x x x
1 2lim f x lim x x 2 x lim x 1
x xx.
2x
1 2lim 1 1
x xx 1
xlim f x ,
1
xlim f x
xlim f x 5 1.
2
2
x x x
xlim f x lim x x 2 x lim
2x 2 x
2x x 2 x
x
x
lim
21
x
x
2
1
21 21 1
x x
.
1 9
52 2
3.278.
2 2
1 3 1 3g x x 9 x 1 x x 9 1
x x xx x, x 0
xlim x
2x
1 3lim 9 1 4
x xx.
4 0 4
xlim g x .
4 0 4
xlim g x .
x
5lim g x
6 4 0 4 .
2 2 2 2
x x xlim g x lim 9x x x 3 4x lim 9x x 3x x 3 x
2
x
9xlim
2x 9x
2
2
x
9x x 3x
23 x
2x 3 x
213
x
xlim
x
2
3
1 39 3 x 1 1
x x
x
2
1 3 1 1lim
x 61 39 3 1 1
x x
.
1 5
16 6
.
3.279.
2 2
2 2x x
1 1lim x x 1 x 1 lim x 1
x x x.
2 2x
1 1lim 1 1
x x x 1
2 2
xlim x x 1 x 1 1
2 2
xlim x x 1 x 1 .
2 2
xlim x x 1 x 1 3 1.
2
2 2
x x
xlim x x 1 x 1 lim
x 1 2x 1
2 2
x x 1 x 1
x
xlim
x
2 2
21 11 1
x x x
. 3 6
2.
3.280.
2
2x x x
1 2lim f x lim x 1 x 2 lim x 1
xx
2x
1 2lim 1 1
xx.
1 0, ,2
2 2
xlim f x .
1
2
2
x xlim f x lim x 1 x 2
2
x
xlim
21 x
22
x 1 x
x
2
1lim 2 2
1x 1 1
x
214
1 2 2
2 2 22 3 3 3
.
3.281.
x x
2lim f x lim x 2 2
x x.
x
2lim 2 2 2 2
x x
22 2 0 ,
2 4 4 2
xlim f x
2 2 0 0,
4
xlim f x .
x
2 1lim f x
2
4.
2
x xlim f x lim 2x 2x 2x
2
x
2xlim
22x 2x
22x 2x 2x
x
2 xlim
x
2
222 2
x
2 2 1 1
2 2 2 6
3.282. f x 24x x 0 x
2 20 16 0 16 1 .
2 2f x x 4 x x 4
x xx x
xlim x ,
2x
lim 4 2x x
.
2 0 2
xlim f x
2 0 2
xlim f x
xlim f x 2 0 2 f x
2 2
2
2
2
x4x x 4x x
f x 4x x 2x4x x 2x
x 4 2x x
.
x x
2
xlim f x lim4
4 2x x
.
215
6 24
4 1 2
16 24 16 576 36 .
24 , 2 36 .
3.283. 4 2 2f x x 4x 3 x x , x .
2
2 4 2x x
4 3lim f x lim x 1
xx x x.
2 4 2x
4 3lim 1 1
xx x x
1
xlim f x , 1
xlim f x .
xlim f x 3 1. :
4
x x
xlim f x lim
2 44x 3 x
2
4 2 2 x
x
x limx 4x 3 x
2
2
34
x
x
2 4
x4 3
1 1x x
2
x
2 4
34
1 xlim xx x4 3
1 1x x
.
2
x
4
34
1 xlimx x4 3
1 1x x
0 ,
xlim f x 0
xlim f x .
xlim f x 3 0 .
4
4 2 2
x x x
xlim f x lim x 4x 3 x lim
2 44x 3 x
4 2 2x 4x 3 x
2
x
x
lim
2
2
34
x
x
2 4
24 3
1 1x x
.
2 3 5
3.284.
2
2 4 2x x
24 6 5lim f x lim x 9
x xx x x.
2 4 2x
24 6 5lim 9 3
x xx x x
216
3
xlim f x 3
xlim f x ,
x
7lim f x
3 3 .
4 3 2 2
x xlim f x lim 9x 24x 6x 5 3x x
4
x
9xlim
3 2 424x 6x 5 9x
4 3 2 2x
9x 24x 6x 5 3x
3
x
2 4
6 524
x xlim xx24 6 5
9 3x x x
3
x
2 4
6 524
x xlim 4x24 6 5
9 3x x x
4 0
xlim f x
4
4 3 2 2
x xlim f x lim 9x 24x 6x 5 3x 4x
24 3 2 2
4 3 2 2x
9x 24x 6x 5 3x 4xlim
9x 24x 6x 5 3x 4x
4
x
9xlim
324x 2 4
6x 5 9x 324x
2
4 3 2 2
16x
9x 24x 6x 5 3x 4x
2
x
x
lim
2
2
510
x
x
2 4
5
324 6 5 49 3
x xx x
.
5 7
43 3
3.285.
2
2 4x x
5 9lim f x lim x 1
x x
2 4x
5 9lim 1 1
x x
1
xlim f x 1
xlim f x .
1
2 4
x xlim f x lim x 5 x 9
4
x
xlim
4x
2 4 x2
4
9 95 lim 5 5
9x x 9x 1 1
x
217
3.286.
x x
xlim x x lim x 1
x.
x 0,
xx 1 1 1 x 1
x x x x x x x.
x x
1 1lim lim 0
x x
x
xlim 0
x.
x
xlim x 1
x
xlim x
x
xlim 1 1 0 1
x.
x x
xlim x x lim x 1
x
x 0 1 x 11 x 1
x x x
x x
1 1lim lim 0
x x,
x
xlim 0
x
x 0 1 2x 11 2x 1
x x x.
x x
1 1lim lim 0
x x
x
2xlim 0
x
1
ux
x u 0 .
2
2
x u 0 u 0
11ux
1 uuxlim lim lim 11x 2 2u 1 u
2u
2
x 4u
x 2
2x x x
x 4 xlim u lim lim
x 2 2x
0
2x u 0
x 4lim lim u 0
x 2
x x x
x x x1
x x x x xlim lim limx 1 1x 1
1x x x
x ,0
xx 1 1 1 x 1
x x x x x x x.
x x
1 1lim lim 0
x x
x
xlim 0
x.
x
x1
1 0xlim 11 1 0
1x
.
3.287.
x x
xlim 2x x lim x 2
x
x
xlim 0
x .
2 2
x x
32
2x 3 xxlim x lim x4x 4 x
x
xlim 0
x .
218
2 2 2x 0 x x 3x1 x 1 1 x 1 1 2 x 3 2 x
x 3 x 3 x 3
2
x
xlim
x 3
2
x
x 2 xlim
x 1
3.288.
x 0
2 2 2
x x x x1 x 1
x 2 x 2 x 2.
2x
xlim 0
x 2
2x
x xlim 0
x 2
2
2
x x
xx x x
lim limx x
x1
x
x
x1
x
,
. .
x x
x xlim lim 0
x x
1
ux
xu 0
2
u 0
lim u u 0
x x
1 1 1lim x x lim x x 1
x x x
1u
x
x u 0 u 0
1 ulim x lim 1
x u
1u
x
x u 0 u 0
1 1lim x lim u 0
x u
u 0,21 1
1 1 u u uu u
u 0
1lim u 0
u.
x 01 1 x 3 x 31 x 1 1 2 x 3 1 x 3
3 2 x 3 2 x
. .
x
x 3lim
2 x
1 x 1, 1 x 1,
2 x x 2 1 3 x x 5
x 101 11
5 3 x x
2 2
2x 2x 1 x 2x 1x 2x 1
5 3 x x.
2
x
x 2x 1lim
3 x x
3.289.
1u
x
x u 0 u 0 u 0
1 1 ulim x 1 lim 1 u lim u 1 1
x u u
1u
x
x u 0 u 0 u 0
1 1 ulim x lim u lim u 0
x uu
1u
x4
4 3x u 0 u 0 u 0
1 u 1 ulim x lim lim
x uu u
219
4 3
x x
1 14x x 4 x
x xlim lim52x 5
2x
,
1u
x3
3 2x u 0 u 0 u 0
1 u 1 ulim x lim lim
x uu u
x x
1 xlim x x 1 x lim
x
1 x
1x
xx 1 x
x
1 1lim x 0
x1x 1 1
x
,
1u
x
x u 0 u 0
1 ulim x lim 1
x u
x 0 x x ,
2 2
x x x x
1 1x x xx 1 x 1 x xlim lim lim lim
xx x x x x1x 1
xx
,
x 0
xx 1 1 x 1
x x x x x x,
x x
1 1lim 0 lim
x x,
x
xlim 0
x
2 2
2x x
1x x
xlim limx 2 x
1
x
x
x
22
1 1 1lim x
x 2221 11 1
xx
,
1u
x
x u 0 u 0
1 ulim x lim 1
x u
2x x
1x x
xlim limx x x
2
2
1x x x
x
x x 2x
x
1 1lim x 1 1 2
x x
2
2 2x x
1
x1 1
xx xlim lim
x x 1 x x x 1 x
x
.
1 1x
x.
220
x
x x
1lim lim 0
x
x 0
1
xlim lim 11
x
.
2 2
2
2x x
x x 1 x x x 1 xlim x x 1 x lim
x x 1 x
2 2
x2
2
x x 1 xlim
1 1x 1 x
x x
x
2
x 1lim
1 1x 1 x
x x
x
x
lim
11
x
x
2
1 11 1
x x
x
2
11
1xlim21 1
1 1x x
.
2x
1
x1
1xlim 21x x 1 x2x
3.291.
2
2
2x x
xx 2x x 2
lim limx x 9
2
2
x 21 2
x x
x
2
19
1x
,
x
xlim 0
x
2
2
4x x
xx 10x x 5 2x
lim limx x
2
2
x 2x1 10 5
x x
x
2
01
xx x
x x
x 2xlim lim 0
x x
3.292.
2x x
x xlim x lim
x 1 2x
x
22
1 xx lim 0
11 x11
xx
2
2
5 3x x
xx x
lim x limx 4x 7x 3
5
11
x
x
3
2 4 5
x4 7 3
1x x x
2x
2 4 5
11
1 xxlim 0 1 0 04 7 3 xx
1x x x
221
4 2
x 1 2xf x
x 3x 2, x .
4 2 4 2
x 1x 1 2xf x 2x
x 3x 2 x 3x 2
4 2
x 11
x 3x 2
x , 1 :
4 2
x 1f x
x 3x 2
4 2
x 1f x
x 3x 2
4 2
x 1
x 3x 2
:
4 2x
x 1lim
x 3x 2
4x
xlim
x
3x
1lim 0
x
4 2x
x 1lim 0
x 3x 2,
xlim f x 0 .
2 2
3
2 3
x x
x x3
3x x x x x xlim lim 03 4x 3x 4
1x x
.
x 0 1 x 11 x 1
x x x
x
xlim 0
x
x 0
2 2 2
1 x 11 x 1
x x x
2x
xlim 0
x
2 2
2x x
1 1 1 2lim 3x 4x 2 lim x 3 4
x x x x
1u
x
x u 0 u 0
1 1lim limu u 0
x x,
2x
2lim 0
x.
2
x
x x5 2
5x xlimx 4
4x
x 0 1 x 11 x 1
x x x
x
xlim 0
x
2x 02 1 x
0 x 1 0x x
2
x
xlim 0
x
3.293.
3
3
3x x x
xx 3 x
lim f x lim lim3x x
3
3
x1 3
x
x
3
1
3x3
x
,
x 0
3 3 3 3 33
x 1 1 1 x 1
x x x x xx
3 3x x
1 1lim lim 0
x x
3x
xlim 0
x
3
3
3x 0 x 0 x 0
xx 3 x
lim f x lim lim3x x
3
3
x1 3
x
x
3
x3
x
.
222
3
x 0lim f x 1 3
x 0
1lim f x
3
1
3
3x 0 x 0 x 0
xx 3 x
lim f x lim lim3x x
2 xx 3
x
x
2
3x
3xx
2
x 0lim f x 3 .
3.294.
x x x x x x
x x x x x x
2 e 2 e 2 e 2 e 2 e 2 ex ln x ln ln ln x ln ln
e e e e e e e e e e e e 1
x x
x xlim 2 0, lim e 0
x
xx
2 elim 1
e e.
x
x
2 eu
e e
x
xx x
2 elim u lim 1
e e
x
xx u 1
2 elim ln limlnu 0
e e
1
x
xx
2 elim x ln 0
e e
3.295.
x 0
4 4 2 4 4 4
4
10 x f x x g x x f x x 1 0 f x
x
4x
1lim 0
x
4
xlim f x 0 4 44 4f x f x f x f x f x . .
xlim f x 0
x 0
2 2 4 4
2
10 x g x x g x x f x x 1 0 g x
x
2x
1lim 0
x
xlim g x 0 g x g x g x . .
xlim g x 0 .
3.296.
f xg x f x g x x
x, x 0 .
x x x
f x g x x xlim lim lim g x 0
x x x
x
xlim 0
x
xx xlim f x x lim g x x x lim x g x 0 ,
x g x g x g x x g x g x
xlim g x 0 .
3.297. x 1 0 x 1 x 0 fA 1,0 0, .
i.
x 0,21 1
1 1 x x xx x
x 0
1lim x 0
x
x ,021 1
1 1 x x xx x
x 0
1lim x 0
x.
x 0
1limx 0
x
223
ii.
1u
x
x u 0 u 0
1 ulim x lim 1
x u
2
x x x
1 1 1 1lim x f x lim x x x lim 1 x 1
x x xx
3.298.
x2x x 23 u
2x x 2x 0 u 1 u 1 u 1
u 13 4 3 3 u 4u 3lim lim lim
3 2 3 3 u 2u 3
u 3
u 1
1
2u 3
x2x 1 x 22 u
x x 2 2x 0 u 1 u 1 u 1
u 12 2 1 2u u 1lim lim lim
3 4 2 1 3u 4u 1
2u 1
u 1
3
23u 1
2x x 2x x
2x 1 x 1 2x xx 1 x 1
4 5 4 4 4 5 4 4lim lim
4 2 4 32 4 4 2 4 4 32.
x4 u x 1 u 4 :
u 4 u 4
u 1 u 4 u 1 1lim lim
4 u 4 u 2 4 u 2 8.
3.299.
x
x x 1
x 1 x 3x x
32 3
lim lim2 3
x
x
23
3
3
x
1
922 27
3
x 1 x
x x 1x
4e 6 2lim
e 2
x x
x xx
4e e 6 2lim
e 2 2
x
x
2
lim
x
x
e4e 6
2
2
x
4e 0 63
0 2e2
2
x
x x 1
x 1 xx x
25 2
lim lim3 5 2
x
x
52
2
2
x2
515 1
2
x
x 2 x 1
x x 2x x
ee 2 3
lim lime 2
x
2
x
x
2 3e 2
e e
e
2
xe
21 4
e
x
2x 1 x
x x 2x x
42 2 1
lim lim3 4 2 1
x x
x
1 12
2 4
4
x x
2
1 1 33 4
2 4
224
xx
x
x xx 1 x x
x 2 x xxx x x2x 2
5 3 5 32 8 2 8 22 24 2 2 5 3 2 2 2
lim lim limee 4 44
ee eee
x
x
3x 1 3x 2 x x
2x 1 3x 2 x x xx x x
x
8125 2 25
1252 5 2 8 25 125lim lim lim
3 2 3 9 4 8 89 3 4
9
x
x
xx
82 25
125125lim
9 83 4
9
x x 1
x 2 xx
4 3 2lim
4 5 3 1
x
x
4
lim
x
x
x
3 21 3
4 4
4
x
x
1 3 0 2 0 1
16 5 0 0 163 116 5
4 4
3.300.
2 2x 1 x 1
x xx 2 x 2 x x
x xlim e e 2e lim e e 2e
2 2x 1 x 1
xx x
xlim e e e
2 21 1
x x x
xlim e e e 2 ,
x ux u
x ulim e lim e
2 21 1
x x
xlim e e 2 2e 2
x
1lim 0
x
3.301. 0 3 :
x
x 3
x
x
x
x
53 1
3 3f x
83 1
3 3
x
3
x
x
x
51
3 3
81
3 3
1
3 :
x
xlim 0
3
3
x
0 1 0lim f x 1 64
0 1 8 0.
225
3 :
3 x x x
x x x
3 3 3 5 26 3 5f x
3 3 8 2 3 8
x
x
x
x
53 26
3
83 2
3
x
x
526
38
23
x
x
x x
526
26 03lim f x lim 13 648 2 0
23
.
3 :
x
x 3
x
x
x
x
3 5
f x3 8
1
x
3
x
x
x
3 5
3 81
3 :
31
x
x
3lim 0 :
x
3
x3
x
x
x x
3 5
lim f x lim3 8
1
.
364 4 .
3.302.
x x
x xx x
2 3lim f x lim
4 2 1.
2
x
x xlim f x lim
x
x
x
2 3
x
x
2 11 4
.
0 2
x
x x
2
lim f x lim
x
x
x
31
2 2
2
x
x
1
414
2 2
2
x
x x x
x x xx x x x
22 2 2 3 2 3
lim f x lim lim lim2 4 2 1 5 2 1
x
x
31
2
2
x
1
1 55
2
3.303. 0 2
x
x x
x xx x
2lim lim
2
x
x
21
x1
21
.
226
2
x
x x
x xx x
22
lim lim2
x
x
12
2
x1
12
2
x x x x
x x x xx x
2 2 2lim lim 0
2 2 2
3.304.
xx
xx x
x x xx
x
1 21 3 2 33 2
f x5
55
2x
23 0
2lim f x
0 5 5.
xx
x
xx
x
1 13 2 3 2
f x
1 51 5
x
13 2 0
lim f x 31 5 0
.
x x2
x x 2
1 23 2 3
3 2f x
1 55 5
2
2x
3 2lim f x
5.
3.305.
3x 3x
3x 3x 3x 3x 3x 3x2xx
2x 2x x 3x
e
e e eeg x ln x ln lne ln lne e e e
e
3x
3x 3x
3xx x
ee
lim lime
3x
3x
1e
e 1
xlim g x ln1 0 ,
0 e
3x
3x
3x 3x3x 3x
3x 3xx x x
e1
e elim lim lim 1
ee e
xlim g x . e
3x 3x 3x
3x
e e 2 eg x ln ln
e 3xe
ln2
xlim g x ln2
227
3.306.
2
2
2
x
2x 3u2x 3
1 xu1 x
x u2x 3lim
1 x
lim e lim e 0
2
x 1u
x 5x
2x x x
x 1 xlim u lim lim
x 5x 2x
0
2
x 1
ux 5x
x u 0lim e lime 1.
2x x 1
ux 2
2
x x
xlim u lim
x
2x x 1
ux 2
x ulim e lim e 0
2
2
x 5u
x 2x,
2
x x
xlim u lim
2x
1
2
2
x 5
ux 2x
x u 1lim e lime e
2x 3 x u
2
2
x x x
xlim u lim x 3 x lim
23 x
2 x
2
3lim 0
3x 3 xx 1 1
x
2x 3 x u
x u 0lim e lime 1
3.307.
1 1uu ux x
1 uu u ux 0x
81 8 1 8lim lim lim
3 83 8
u
u
8 1
8
u1
38 1
3.308.
x 0 x 0 x 0 x 0
lnx
lnx lnx lnx lnxlim lim lim limx xln x ln x lnx lnx
ln lnx ln lnxx x
lnx
x 0 x 0
lnx 1 1lim lim 1
1 xln x 0 0 1ln 1
lnx x
x
ux
x 0 x 0
xlim u lim 1
x
u 1x 0
xlim ln limlnu 0
x
x 0
1lim 0
lnx
x 0
lim lnx
3.309.
lnx u
3 2 3 2u u ux 0
3lnx 2 3u 2 3ulim lim lim
4ln x ln x 1 4u u 1 34u
2
0
2
2x x x
x xlim lnx ln x 2x 3 lim ln lim ln
x 2x 3x
2
ln1 02 3
1x x
x 2
x 2 x 2 x 2
x 2x x x
e 3lim ln e 3 x 2 lim ln e 3 lne lim ln
e
x 2
x
e
lim ln
x 2
x 2
31
e
e
ln1 0
228
2
2
2x x u 1
x 1lim 2ln x 1 ln x x 1 lim ln limlnu 0
x x 1
22 2
2 2x x x x
x 1 x 2x 1 xlim u lim lim lim
x x 1 x x 1 2x
1
3
2
2x x u
x 2lim 3ln(x 2) ln(x 3) lim ln lim lnu
x 3
33 2 3
2 2x x x x
x 2 x 6x 12x 8 xlim u lim lim lim
x 3 x 3 2x
3
3
3x x u 8
8xlim 3ln 2x ln x 1 lim ln limlnu ln8
x 1,
3 3
3x x x
8x 8 xlim u lim lim
x 1 3x
8
3.310.
2
2
2x x x
2x x 2 xlim ln 2x x 2lnx lim ln lim ln
x 2x
2
2u 0
xlim lnu
x
22
2 2 2
x2 x 2 2 1u
x x xx x x
x 0 2 2 2
2 1 2 1 2 1u u
x x xx x x.
2x
2 1lim 0
x x
xlim u 0 u 0
3.311. 2g x f x x x 2 x 1 1
xlim g x 4 .
x 2f x g x x x 2 x 1
22
2
x x 2 x 1g x
x x 2 x 1
2 2
2
x x 2 x 2x 1g x
1 2 1x 1 1
x xx
x
g x
13
x
x
2
1 2 11 1
x xx
2
x x
13
xlim f x lim g x1 2 1
1 1x xx
3 5
42 2
.
f : f x f x x .
:
x h
xlim f x
h h
5lim f h lim f h
2.
3.312. 2 2f x x 2x 3 x g x f x g x x 2x 3 x ,
xlim g x 4
229
2
x x
xlim f x lim g x
22x 3 x
2 x
x
lim g xx 2x 3 x
32
x
x
2
4 1 52 3
1 1x x
x u
x x u ulim f x lim f x lim f u 5
3.313.
2
2g x x x 2x 3
3f x x 2x 3 x g x f x3
2
x x
1 xlim f x lim g x
3
2x
2 x
x2x 3 1
lim g x3x x 2x 3
32
x
x
2
1
32 31 1
x x
3.314.
2x 12
g x2 x 28x 1 f x 3x g x f x 3
x 8x 1 8x 1 x
x
g xlim 0
8x 1
2u2 x 2
x u 0 u 0 u 0
4
x 2 4 u 1ulim lim u lim168x 1 x 16 u u 4
1u
,
x
3lim f x
4
3.315.
x 02
2
1f x 2x
1x g x f x g x x 2x 3 x 2xxx 2x 3 x
.
2
2
x x
xlim g x x 2x 3 x lim g x
22x 3 x
2x 2x 3 x
x
x
lim g x
32
x
x
2
32 3
1 1x x
1u
x
x u 0 u 0
1 ulim x lim 1
x u
xlim f x 1
3.316.
3
x x x x
x x x xlim f x x lim x lim x lim x x 1
x 1 x 1 x 1
230
2
x
xlim
x
x
11
x
x
x
1 xlim
x x 1
1
11 x
x
1
211 1
x
3
x x x x
x x x x x xlim f x x lim lim lim x
x 1 2 2x 1x
1
211
x
3.317.
2
x
x
lim
2
2
f x 63
x x
x
2
4 52f x 6x
x x
2
x
2
f x 63
x xlim4 5
2 f x 3xx x
3 3 00
2 6 0 0
3.318.
2
2
2 3x x
x3x xf x 5 4x
lim limx f x xf x 2x
2
2
f x 5 43
x xx
x
3 33
3 1f xf x 2x
x
3.319.
2
2
2x x
xf x3 x 2 f x 4f x 5x
lim limxf x x 3 f x 2
2
2
2 4 53 1
x xf x f x
xf x
2
33 1 2
1 1x f x xf x
3.320. 1 : 5 5 4 5
f x f x 32x f x f x 1 32x .
x 0 4f x 1 0 , f x 0 1
5 5f x 32x 0 f x 2x x 0 ,
5 4
f x 20
x x
4x
2lim 0
x
5x
f xlim 0
x.
3.321.
2
2 2
x 1 x 1x 1 f x x 1 f x
x 1 x 1.
2 2x x x
x 1 x 1 xlim lim lim
x 1 x 1 2x
0
xlim f x 0 .
3.322.
2 2
4 2
4 4
x x x x1 x f x x x f x
1 x 1 x.
2 2
4x x
x x xlim lim
1 x 4x
2
0
2 2
4x x
x x xlim lim
1 x 4x
2
0
xlim f x 0 .
3.323. 2 1
f x g x 2f x g x 2 f x g x 2x
231
2 2 1
f x 1 g x 1x
2 2
f x 1 g x 1 0
x
1lim 0
x
2 2 2 2
xlim f x 1 g x 1 0 1 1 0 1 .
3.324.
xu
x 1
x x
xlim u lim 1
x
x xu u
uxu x u x u 1 u x
u 1, u 1
22 2
2
2x u 1 u 1
xf 2
f u 2 u u 1 f u 2ux 1lim x 1 lim 1 lim
u 6u 5u 56x 5 u 1 u 16 5u 1 u 1
2u 1
f u 22u 1lim
u 5u 1
u 1
u 1
f u 22u 1 1lim 6 1
u 5 u 1 6
3.325. P x -
1
2x
P xlim 0
x 1 2
2x
P xlim
x 1.
2x
P xlim 1
x 1 2 .
2P x x x , 0
2 2
2 2 2x x x
P x x x xlim lim lim 1
x 1 x 1 x
2 x 12 2
2
x xf x x x f x x 1
x 1
2 2
x 1 x 1lim x x lim f x x 1 1 0 1
2
2 2x 1 x 1 x 1
x 1P x x x 1lim lim lim
x 1 x 1
x 1 x 1
x 1
23 4
2x 1 5
2P x x 4x 5 .
3.326. f x - 1
x
f xlim
x 1.
f x x , 0
x
f xlim 4 4
x, f 0 5 5 .
x 4
4x 5lim 7
x 4
x 4
4x 5lim
x 4
4
x 4
4x 5 21lim 7 1
x 4
232
3.327.
2 2 2 2 2 2f x 5g x f x g x f x g x
5f x g x f x g x f x g x f x g x
2 2f x 5g x
f x g x f x g xf x g x
2 2
x
f x 5g xlim 0
f x g x
3.328. y 0
x 0
2 2 2 2
x xf x f x1 1f x x x f x x
x xx xx x
x
1lim 0
x
2x
f xlim 0
x
2
2
2x x
x6x 3f x
lim lim2x f x
2
2
f x6 3
x
x
2
3f x
2x
3.329. K x x 30.000
x x
x 30.000lim P x lim 1
x
x 0 x 0
x 30.000lim P x lim
x
3.330. A ,0 , M
x 3
2
3
2 2 2AB: y 0 x y x
3 3 3
x 0
2y
3
2B 0,
3,
2
OAB3
2 2
x x x
x xlim OAB lim lim
x 3 x
2
x 3 x 3
xlim OAB lim
x 3
3.331. 2 2B x ,
2 2B x
AB x,
2 2
x x x
xB x
lim lim limAB x
2
21
x
x1
3.332.
xx 1 1 x 1
x x x x x x x 0,
1 x 1
x x x
233
x
1lim 0
x
x
1lim 0
x,
x
xlim 0
x
2 2 2
2
1x f x g x 2 f x g x x x 1
x
x 0 :
2 2
2 2 4 2
2f x 2g x x 1 1f x g x
xx x x x
2 2
2 2 4 2x x
2f x 2g x x 1 1lim f x g x lim
xx x x x
2 20 0 ,
2 2x x
2f x 1lim lim 2f x 0
x x
2 2x x
2g x 1lim lim 2g x 0
x x.
3.333.
22
2 2 2
2y y2
2
2xy 2 x x
y2y x xy 2x ylim lim
xy 1 1y x
y
2 x x 0
x 0
2 x x
f xx
x 0 .
x 1 1 x 1
x x x x x
x x
1 1lim lim 0
x x
x
xlim 0
x
x x
xlim f x lim 2 1 1
x
3.334. z x 1 i x 1 i x x x x i
2 2 2 22 2
2 2 2
1 2
z x x x x 2x 2 x xf x 1
xz z x 2x 2x
x 1 1 x 1
x x x x x
x x
1 1lim lim 0
x x
x
xlim 0
x.
2
x x
xlim f x lim 1 1
x
2
x 0 x 0
xlim f x lim 1 2
x
2
1u 2
x
x x u 0 u 0
1
1 uxlim f lim 1 lim 1 21x u
x
234
3.335.
2 2 2 2
2 2 2 2
x y 2 4y 3lim f lim
x y 2 4y 3
2 2 2 2
2 2
2
x y 4y 1 x y 4ylim
2x y 2 4y1 1 3
2 2
22x y 4y0 x y 2 4
2
0,2 2 .
2 2 2 2lim g lim 2x 3x 2y 5y
2 2
2 2 2 2
2x 3x 2y 5ylim x y
2x 3x 2y 5y
x y 2 x y 2 0 .
0 2 2
d , 0 25
3.336. f 1 2
x ,x 1 2
x x
1 2
f x f x 1 2
f f x f f x 1 2
f x f x
1 1 2 2 1 2
f f x f x f f x f x x 5 x 5 1 2
x x
f
f 1 1
1f x y 1
x f y 1 19f y 6y f y 5 f y 6y 9f y 5
1f x 6x 9f x 5 , x
i.
2
x x
f f x f f x f xlim lim k
x f x x
f x u
x u u
f f x f ulim lim k
f x u
ii.
x 0 f f x f x 9
9 f f x 6f x x 5 9 6 1x x x
2 2
x x
f f x f x 5lim 9 lim 6 1 9k 6k 1 9k 6k 1 0
x x x
2 1
3k 1 0 k3
3.337. x1 e 0 x 0 x
1 e 0 A ,0 .
x x x x x x1 e 1 e ln 1 e ln 1 e ln 1 e ln 1 e 0 f x 0
1 2
x ,x 1 2
x x 1 2 1 2 1 2x x x x x xe e e e 1 e 1 e
1 2x xln 1 e ln 1 e 1 2x x
ln 1 e ln 1 e 1 2
f x f x f
1 1
235
x y
x x
x y
1 e 1 ef x y ln 1 e ln 1 e y ln y ... x ln
1 e 1 e,
x
1
x
1 ef x ln
1 e
i.
4 3 2 4
3 2x x
f 4 x 3x 7x 6x 2 f 4 xlim lim
x 5x 9 3x
ii.
x x
1x x
1 e 1 eln ln
f x f x 1 e 1 e
x xlim e e lim e e 0
3.338. x 0
3 3f x f x x
2 3x x x
3 3
3
x 0 x 0
f x f x xlim 2 lim 3 k 2k 3
x x x 1
x 0
3 3f x f x x
2 3x x x
3 3
3
x 0 x 0
f x f x xlim 2 lim 3 k 2k 3
x x x 2
1 , 2 3 3k 2k 3 k 2k 3 0 k 1
3 3
3
x x
f x f x xlim 2 lim 3 2 0
x x x,
3 3
3
x x
f x f x xlim 2 lim 3 2 0
x x x
3 22 0 2 0 0
x
xlim 0
x
i.
x 0f x
g x f x xg xx
x 0 x 0limf x lim xg x 0
2 x u
x 2 x 2 u 0 u 0limh x lim f 2 x 2 lim f u 2 2
ii.
x 2lim xh x 2 2 xh x 2 0
x 2lim 6 2x 2h x 2 ,
6 2x 2h x 0
x 2 x 2
xh x 2 6 2x 2h x xh x 2 6 2x 2h xlim lim
x 2 x 2
x 2
h x x 2lim
2 x 2
x 2 0
236
3.339. f x x 2014 2014 f x x 2014 x 2014 f x x 2014 1
x 0
f x2014 20141 1
x x x
x
f xlim 1
x
1 x 0
1 1
f x1 2014 1 2014
x x x x x.
x
f xlim 0
x 1
x
f xlim 1
x
2014u
x
u ux 0 x 0
20142014f
2014f u2014 xlim xf lim lim 2014
2014x u
x
2
2 2
2 2x x
xf
x 3x 5x 6 xlim f lim x 2
x 3x x
x 3
,
2
2
xu
x 3
2x u u
xf
f ux 3lim lim 1
ux
x 3
3.340. 1 2
x ,x 1 2
x x 1 2
f x f x 3 3
1 2f x f x
3 3
1 1 2 2 1 2 1 2f x 3f x f x 3f x x 4 x 4 x x
f 1 1
1 3f x y ... f x x 3x 4 , x
3f x 3 3 3 3 x 4 x 32 f 32 3
2 2 2
f
x 4x x 4x x 4x 5 2f 2 3 f 2 f 32 2 32 2 x 4x 5 x 5,1
1 3
3 3x x
f x x x 3x 4 xlim lim 0
x xx x
3
3x
x 3x 4lim 1
x
x
xlim 0
x
237
3.356
x 0
x 0 x 0
lim f x lim e e 1,
x 0 x 0
lim f x lim x 1 1 0f 0 e 1.
x 0 x 0
lim f x lim f x f 0 1 0
x 1.
0
2 0
x 1 x 1
x 3 2limf x lim
x 1
2 2
x 1 2
x 3 2 x 3 2lim
x 1 x 3 2
2
x 1 2
x 3 4lim
x 1 x 3 2
x 1
x 1lim
x 1
x 1
2x 3 2
2x 1
x 1 1lim
2x 3 2,
x 1
1f 1 2 limf x
2f
0x 1.
x 1 x 1
x 1lim f x lim
x 1
x 1 2 ,
x 1 x 1
lim f x lim 3x 1 2 f 1 2
x 0 x 0 x 0 x 0
1 x 1 1 x 1 x 1 1lim f x lim lim lim
x x 2x 1 x 1 1 x 1
x 0 x 0
x 1lim f x lim
2x 2,
1f 0
2f
0x 0 .
3.357. f ( , 1)
( 1, )
f 0
x 1 :
x 1 x 1
1lim f x lim 1
x,
2
x 1 x 1lim f x lim x 2x 1
1f 1 1
1
x 1 x 1lim f x lim f x f 1 ,
f 0
x 1.
f
x 1
2 x 3 x 1x 2x 3f x
x 1 x 1 x 3 ,
x 3 , x 1f x
2 , x 1.
,1 1, f
.
f 0
x 1
x 1 x 1lim f x lim x 3 4 f 1 ,
f 1 2 , f 0
x 1.
2y x 2x
1
yx
238
3.358. x 1 f 0
x 1
x 1 x 1 x 1 x 1
2 x 1x 1 x 2 5 x 1 x 2 5limf x lim lim lim
x 1 x 1 x 1 2 f 1
f
x 0 f x 0
2 2 2 2 2 21 1 1f x x x x x x x
x x x.
2 2
x 0 x 0
lim x lim x 0
x 0
lim f x 0 .
x 0 x 0
xlim f x lim 1 0
x, f 0 0 f
3.359. x 0 2 2
1 1g x f x f x f x f x g x f x
x x.
x 0lim f x f 0 0
x 0lim f x f 0 0
x 0limg x 0 g 0
g 0
x 0 .
3.360. f 2
x x .
f 22x 1, lnx , 3x .
f 3 2x x 3 2
x 3x 4 .
f 2x x 2 , x , 3x .
f x ,
2x 1
x 4.
f x , 4
x 22x 3 .
3.361. x 3 ,
2 2 2
x 3 x 3
1lim f x lim x 3
3
x 3 ,
x 3 x 3lim f x lim x 5 3 5
2f 3 3 f
0x 3 :
x 3x 3lim f x lim f x f 3 .
23 3 5 2
3 2 0 1 2
3.362.
3
2
x 0 x 0 x 0
1x
x 1xlim f x lim lim xx x x
2
x 0
1 1lim x
x x
x
2 2 2 21 1x x x x
x x 2 2 21
x x xx
.
2
x 0lim x 0 ,
2
x 0lim x 0
2
x 0
1lim x 0
x.
x 0
1lim f x 0 0
1.
f 0
x 0 :
x 0lim f x f 0 0 .
239
3.363. 1
ux
x 0 ux 0 u .
u uu ux 0
1
1 1ulim f x lim lim 0u1 e 1 e
,
u uu ux 0
1
1 1ulim f x lim lim 0 0 0u1 e 1 e
f 0 0 f 0
x 0 x 0 f
f
3.364. x 1,2 f
1
x 1 2
x 2 . x 1 x 1 0 , x 1 x 1
2
x 1 x 1
2 x 1 x 3x 2lim f x lim
x 1
x 1
2 x 1 x 1 x 2lim
x 1
x 1
x 1lim
2 x 2
x 1
x 1
lim 2 x 2 1
2
x 1 x 1
lim f x lim x x 1 3 f 1 .
3 1 2 1 .
2
x 2
2
x 2 x 2
lim f x lim x x 1 7
x 2 x 2
lim f x lim 3 x 2 6 2 f 2 .
7 6 2 7 7 2 1 2
9
2 8
77 7
8
.
3.365.
2
x 1 x 1 x 1
x x 1x x xlim f x lim lim
x 1
x 1
x 1 ,
x 1 x 1 x 1
x x 1 x 1 x 1 xlim f x lim lim
x 1 x 1
x 1
x 1lim
x
x 1
2x 1
23
122
3.366. f 0
x 2
x 2 x 2
lim f x lim f x f 2 .
x 2
2
2x xf x f x x 2 x x
x 2
2
x 2 x 2
lim f x x 2 lim x x 0 6 6 .
2
x 2 x 2 x 2
x 3 x 2x x 6lim f x lim lim
x 2 x 2 5 .
240
5 6 5 11.
2
x 2 x 2 x 2
x 24x 11x 6lim f x lim lim
x 2
4x 3
x 2 5
3.367.
2
2x 3 x 3 x 3
x 3x 3 x 6x 9lim f x lim lim
x 9
2
x 3
x 3 x 3
1
6
x 3 x 3
x 3 3lim f x lim f 3
5x 15 30 30
3 1 53 5 5 3 0
30 6 3 1 2 .
3.368. x 0 f
2
2
1 1 1u x x
ux u
x 0 u
u
ux 0
1lim f x lim e 0 0 0
u
x 0 f
2
x
xx 0 x 0
ln x 1lim f x lim e x 0 1 1 f 0
e.
1 0 1
3.369. x 1 1 x 2 f
x 2 f
x
x 1 x 1
lim f x lim e e f 1 ,
x 1 x 1
lim f x lim x 1 lnx 0 ,
e 0 1 .
x 2 x 2
lim f x lim x 1 lnx 1 ln2 f 2
1u
x 2
1 uu ux 2 x 2x 2
1 1lim f x lim lim 0
1 e1 e
,
1
1 ln2 0ln2
e
1ln2
3.370.
2
x 2 x 2 x 2
x 4 x 2 2x 5 3x 2x 8lim f x lim lim
2x 5 92x 5 3
x 2
x 4 x 2lim
2x 5 3
2 x 218
2
x 2 x 2
lim f x lim x 1 x 4 2 2 f 2
241
f 4 2 2 18 2 10 1
2 2 2g x h x 1 x x 3 x 2 x x 2 x 2 0 2
gC ,
hC 2
2 2 2
0 2 8 0 4 4 8 0 2 24 4 0
2
2 0 2 1 5 10 2 2 2 4 .
3.371.
x 1 x 1
lim f x lim f x f 1
2 3 3 2 3
x 1 x 1
lim 3x ( 1)x 1 lim 2x ( )x 3
3 2 3 23 2 1 1
2
2 2 1f 2 15 16 2 2 15
2
2
3 2 2 11 1
2 3 2
2 2 22 2 2 3
1 2 3 0
21 2 2 3 0 1 2
2 2 3 0
1
1 2 1
02
3.372.
2 2 2 2
x 1 x 1
lim f x lim 1 x 2 x 12 1 2 12 11
x 1 x 1
lim f x lim x 1 1
1 11 12 211 11 144
3.373. 0
x 1,
x 1 x 1
lim f x lim f x f 1
2 2 2 3 2
x 1 x 1
lim x 4 x lim x x 4 1 4
2 2 21 4 1 4 2 2
4 4 2 0 2 2
2 2 0
2 0 2 0 .
3.374. x 0 g . g
0
x 0
2 2
x 0 x 0limg x g 0 lim 3 x x 4 1 4 1 3 1
12
x 1 x 1
lim f x lim x 2 x 5 2 5 3 2 5 8 3
1
x 1 x 1
lim f x lim 2 x 2 2 2 6 2 2 8 3 f 1
x 1 x 1
lim f x lim f x f 1
3.375. x 2 :
2
2x 2x 7x 10
x 2 f x x 7x 10 f xx 2
x 5
x 2 x 5
x 2 x 2limf x lim x 5 3 .
0
x 2
x 2
f 2 limf x 3 .
242
3.376. x 1
2
2 x 3 2x 1 f x x 3 2 f x
x 1.
0
x 1.
:
2
x 1 x 1
x 3 2f 1 limf x lim
x 1
2
x 1 2
x 3 4lim
x 1 x 3 2
x 1
x 1lim
x 1
x 1
2x 3 2
2x 1
x 1 1lim
2x 3 2.
2x 3 2
, x 1x 1f x
1, x 1
2
3.377.
3 2
2
x 2x f x x 4xg x
x 9 3 3 2 2x 2x f x x 4x g x x 9 3
2
3 2 3 2
g x x 9 3x 4x
f xx 2x x 2x
, x 0 x 2
2
3 2 3 2x 0 x 0
g x x 9 3x 4x
f 0 lim f x limx 2x x 2x
2
x 0
g x xlim
2x
2
4 x
x 2 x 9 3 x
4x 362 1 5
4x 2 6x 2
3.378.
x xf xg x x xf x g x x x
x x
x xf x g x 1
x x, x 0
x 0 x 0
x xlim f x lim g x 1 1 2 1 1 3 f 0
x x
3.379.
3 2 2xxf x 5x 2 x x f x 2 x 5x
x, x 0
2
x 0 x 0
xlim f x lim 2 x 5x 2 f 0
x
3.380.
2
2
1
1x
x
f x x
g x f x xg x xx
, x 0
x 0
2 2 21 1
x x1 1 x x x .
243
2 2
x 0 x 0limx lim x 0
2
x 0
1
xlim x 0 .
2
x 0 x 0
1
xf 0 lim f x lim xg x x 0
3.381.
x 2 f x x 2 x 2x 2 2g x f x g x
x 2 x 2x 2 2, x 2 , x 2 .
x 2 x 2
x 2x 2 2lim f x lim g x
x 2 x 2
x 2
x 2lim g x
x 2
x 22 f 2
x 2x 2 2
x 2 u
x 2 u 0 u 0
x 2 ulim lim 1
x 2 u
3.382.
f x x 2g x f x g x x 2 x 2
x 2, x 2,2 2,
x 2
f 2 lim g x x 2 x 2 2
x 2 x 2 x 2
f x f 2 g x x 2 x 2 2 x 2 4lim lim lim g x
x 2 x 2 x 2 x 2 2
x 2
x 2lim g x
x 2
5
4x 2 2
3.383.
x 1 u
x 1 u 2u 2
f x 1 f ulim 3 lim 3
x 1 u 2
x 2
f xlim 3
x 2.
f xg x f x g x x 2
x 2, x 2 .
x 2f 2 limg x x 2 0 .
x 2 x 2
g x x 2f x f 2lim lim
x 2 x 2 3
3.384.
2f x x 2x 5
g xx 2
, x 2
x 2lim g x 2 .
2 2f x x 2x 5 g x x 2 f x g x x 2 x 2x 5 .
2
x 2 x 2lim f x lim g x x 2 x 2x 5 2 0 4 4 5 3 .
fC M 2, 3 f( 2) 3 .
x 2lim f x f 2 3
0x 2 .
244
3.385. 3 h x h x 3 h 0 : x 3 0 x 3 ,
h 0 x 3
f 3 h f xlim lim 2
h x 3.
f xg x f x g x x 3
x 3,
x 3lim g x 2 ,
x 3 x 3
lim f x lim g x x 3 0 .
f 0
x 3
x 3
f 3 lim f x 0 .
3.386.
x 1 u
u 1u 1x 0
f x 1 f ulim 4 lim 4
x u 1
x 1
f xlim 4
x 1.
f xg x f x g x x 1
x 1, x 1.
x 1 x 1f 1 limf x limg x x 1 0 .
x 1 x 1
g x x 1f x f 1lim lim
x 1 x 1 4
3.387. x 0 f 0 f 2 3 2 f 2 3 f 2 1
2 2
x 0 x 0 x 0 x 0limf x 2 lim x 3 f x lim x 3 limf x 3 2 1
x 2 u x u 2 x 0 u 2 .
x 0 u 2limf x 2 1 limf u 1.
u 2limf u f 2 1,
1x 2 .
3.388. x 0 f 0 f 1 8 f 1 5 .
2
x 1 h 0 h 0limf x limf 1 h lim 4h 4h 8 f h 8 3 5 f 1 .
3.389. x 3 2 2
f x 3 f x 2 x 3 6 x 3 19 2x 18x 55
x 0 f 3 f 0 55 f 3 50 .
x 3 h2
x 3 h 0 h 0 h 0lim f x limf 3 h lim 2h 18h 55 f h 50 f 3
3.390.
f x
g x f x xg xx
, x 0 .
x 0 x 0
f 0 limf x limxg x 0
x 1 h
x 1 h 0 h 0 h 0limf x limf 1 h limf h 0 x 0 f 1 f 0 0
3.391.
x 3 u
x 3 u 6u 6
f x 3 f ulim 2 lim 2
x 3 u 6
x 6
f xlim 2
x 6.
f xg x f x g x x 6
x 6, x 6 .
x 6 x 6
f 6 limf x lim g x x 6 0 ,
x 6 x 6
g x x 6f x f 6lim lim
x 6 x 6 2 .
245
3.392.
x 0limf x f 0
0
0x xlim f x f x ,
0x .
0
x x h 0
x x h 0
0
0 0 0 0 0x x h 0 h 0lim f x limf x h lim f x 2f h f x 2f 0 f x 0 f x
3.393.
x 3lim f x f 3
0
0x xlim f x f x ,
0x .
0
x x h 3 0
x x h 3
0
0 0x x h 3 h 3lim f x limf x 3 h lim f x 3 f h
0 0 0f x 3 f 3 f x 3 3 f x .
3.394. x y 0 f 0 f 0 2f 0 f 0 0 0
x 0 ,
x 0limf x f 0 0
0
0x xlim f x f x ,
0x .
0
x x h 0
x x h 0 .
0
0x x h 0lim f x limf x h
0 0h 0lim f x 2f h x h
0 0 0 0 0
h 0 h 0f x 2limf h limx h f x 2 0 x 0 f x
3.395. x y 1 f 1 1. 1
x 1
x 1limf x f 1 1.
0
0x xlim f x f x ,
0x .
0
0
xh x x h
x
0x x h 1.
0
1
0 0 0x x h 1 h 1 h 1lim f x limf x h lim f x f h f x limf h
0 0 0
f x f 1 f x 1 f x
3.396. x y 0 f 0 f 0 f 0 f 0 1 0
x 0
x 0limf x f 0 0
0
0x xlim f x f x ,
0x .
0
x x h 0
x x h 0 .
0
0x x h 0lim f x limf x h
0h 0lim f x f h
0 0h 0
f x limf h f x
xlim f x f
0
0x xlim f x f x ,
0x .
0
x x h 0
x x h .
0
0 0x x h hlim f x lim f x h lim f x f h
0 0 0
f x f f x f x .
3.397. x y 0 f 0 f 0 f 0 f 0 1 0
x 0
x 0limf x f 0 0
0
0x xlim f x f x ,
0x .
0
x x h 0
x x h 0 .
0
0x x h 0lim f x limf x h
0h 0lim f x f h h h
0 0h 0
f x lim f h h h f x
246
3.398.
xx 0
x xf x 3xe
g x f x 3xe xg x f x xg x 3xex
x
x 0 x 0limf x lim xg x 3xe 0 x y 0 f 0 f 0 f 0 f 0 0 .
0
x x h 0
x x h 0 .
0
0x x h 0lim f x limf x h
0xh
0 0 0h 0lim f x e f h e 3x h f x
3.399. x 0
2 3xx 2x f x 5x x 1
x.
2
x 0
xlim x 2x 1
x,
3
x 0
lim 5x x 1 1
x 0
lim f x 1.
f 0
x 0 :
x 0 x 0
f 0 lim f x lim f x 1.
3.400. 2xf x x x x 2 2
x xf x x x x 2 2x x x xf x x x x
x 0 , 1 : x x f x x x
:
x 0lim x x 0
x 0lim x x 0 .
:
x 0lim f x 0 .
f 0
x 0
x 0 x 0
f 0 lim f x lim f x 0 .
3.401. x 3 18 f x 18 f 3 18
x 3lim6x 18 ,
2
x 3lim x 9 18
x 3lim f x 18 f 3 .
1u
x2
2x 1/3 u 3 u 3
1 1lim x f lim f u
x u.
x 0
2
2 2
6 1 96x f x x 9 f x 1
x x x.
x 3
6lim 2
x,
2x 3
9lim 1 2
x
2x 3
1lim f x 2
x
3.402.
x 0
2 4 4 2 4x f(x) x 1 x x x f(x) x 1 x
4 4
2 2 2 2
x 1 x x 1 xf x
x x x x
24 2
2
2 2 2x 0 x 0
x 1 x x 1 xlim lim x
xx x x 1 x
2 2
2
x 0
x x 1 1lim x
x x 1 x 2
4
2 2x 0
x 1 x 1lim
2x x
x 0
1lim f x
2.
f 1
f 02
.
247
3.403.
x 0
2 10x 1x x 10 10 xf x 10 x
x2 10
2
9x x 10 10 1 1f x x
x x2 10
x 0
9 9 91 11 1 x x x
x x
9
x 0
1limx 0
x,
9
x 0
1 1 10lim x
x 202 10.
2 2
x 0 x 0
x x 10 10 x x 10lim lim
x
10
2x x x 10 10
x 0
xlim
2x 1
x
2
10
20x x 10 10
x 0
10lim f x
20f
10f 0
20.
3.404. 2 2 22
f x x 1 x 2x 1 x 1 x 1 f x x 1 x 1
2 2
x 1 x 1 f x x 1 x 1
x 1 0 f 1 0 f 1 0
x 1limf x 0 .
x 1
x 1 f x f 1 x 1x 1 x 1
x 1 x 1 x 1
x 1 u
x 1 u 0 u 0
x 1 ulim x 1 lim u 1
x 1 u
x 1
x 1lim x 1 1
x 1,
x 1
f x f 1lim 1
x 1
x 1
f x f 1lim 1
x 1
3.405. 22 2 2
f x 2f x 1 x f x 1 x x 0 :
2
f 0 1 0 f 0 1 0 f 0 1 f x 1 x 1 x f x 1 x .
x 0lim f x 1
0x 0 .
3.406. x 1 f 1 g 1 0 x 2 f 2 g 2 0 .
2 2 20 f x f x g x
2
x 1 x 1limf x 0 limf x 0 f 1
2
x 2 x 2limf x 0 limf x 0 f 2
3.407. x 0 2 2 2f 0 g 0 0 0 2 2
f 0 g 0 0 2 2f 0 g 0 0
f 0 g 0 0 2 2 20 f x g x x
2
x 0lim x 0
2 2
x 0lim f x g x 0 .
248
2 2 20 f x f x g x
2 2
x 0lim f x g x 0
2
x 0 x 0limf x 0 limf x 0 f 0
0x 0 .
x 0limg x 0
0x 0 .
3.408. x 0 2 2f 0 g 0 2f 0 g 0 2 2
f 0 2f 0 g 0 g 0 0
2
f 0 g 0 0 2
f 0 g 0 0 2
f 0 g 0 0 f 0 g 0 .
2 2 2f x g x 2f x g x x
2 2 2f x 2f x g x g x x
2 2f x g x x .
2
x 0limx 0
2
x 0lim f x g x 0 .
2
f x g x 0
2
x 0lim f x g x 0
2
x 0 x 0lim f x g x 0 lim f x g x 0 .
h x f x g x f x h x g x
x 0limh x 0 .
x 0 x 0 x 0 x 0limf x lim h x g x limh x limg x g 0 f 0 ,
0
x 0 .
3.409. 22 2 2 2
f x 4g x 4f x g x x 0 f x 2g x x
x 0 2 2
0 f 0 2g 0 0 f 0 2g 0 0 f 0 2g 0
2
x 0lim x 0
2
x 0lim f x 2g x 0
2 2
f x 2g x f x 2g x f x 2g x .
.
x 0lim f x 2g x 0
f x 2g x h x f x 2g x h x
x 0 x 0limf x lim 2g x h x 0
3.410. x 0
x
f x 2x 5x
x 0 x 0
xlim f x lim 2x 5 f 0 5
x
x 0
x
f x 2x 5x
x 0 x 0
xlim f x lim 2x 5 f 0 5
x.
f 0 5
3.411. x 2
3 x 2x 6x 4f x
x 2
2x 2x 2
x 2 2
x 2x 2
x 2
3
2x 6x 4f x x 2x 2
x 2x 2 2
f x x 2x 2
2
x 2 x 2f 2 limf x lim x 2x 2 6
249
3.412. 3 2 2 2f x 2f x x f x f x 2 x 2
f x 2 0 x ,
2
2
xf x
f x 2x .
2 2 2
2 2
x x xf x
2f x 2 f x 2 2f x 2 2 x .
2 2 2
x x xf x f x
2 2 2.
2
x 0
xlim 0
2
2
x 0
xlim 0
2
x 0lim f x 0 . x 0 ,
3 2f 0 2f 0 0 f 0 f 0 2 0 f 0 0 2
f 0 2 0
x 0limf x f 0 0 f
0x 0 .
3.413. 3f x f x 3x 3
0 0 0f x f x 3x
3 3 0
0 0 0 0 2 2
0 0
x xf x f x f x f x x x f x f x
f x f x f x f x 1
0
0 02 2
0 0
x xf x f x x x
f x f x f x f x 1,
2 2
0 0f x f x f x f x 0 0
0 0 0x x f x f x x x
.
0
0x xlim f x f x 0
0
0x xlim f x f x
3.414.
2 2 2 2g x x
0 g x x g x x 0 g x x 0 g x xg x x
g x x x g x x
x 0limg x 0
3.415.
xlim f x
12 2u
x
x x x u 0 u 0
u1 1 1x x x
ux x xlim lim lim lim22x 2 2u
xxx
.
0
x
1lim f x .
0
x u 0 u 0
u
u 1ulim f x lim lim2u u 2u
u 0
ulim 1
u
u 0
1lim
2u u 0 x .
1, 0
g
0 , 0
,
0 0
1lim g lim g x 0
0, .
250
3.416. x y 1 f 1 2f 1 f 1 0 0
x 1,
x 1limf x f 1 0 0,
0
x 0,
0
0x xlim f x f x .
0
0
x hx x h
x 1.
0x x h 1.
0
2 2
0 0 0 0x x h 1 h 1lim f x limf x h lim f x f h x x h h
0 0
h 1f x limf h 0 f x
0
2 2
0 00 0 0
x x h 1 h 10 0 0 0
f h x x h hf x f x f x h f xlim lim lim
x x x h x x h 1
0 0
0 0h 1 h 1
0 0 0 0
f h x x 1 h 1 h1 1 4lim lim 4 x 1 x 1
x h 1 x h 1 x x
3.417. x 0 22
f 0 4f 0 4 0 f 0 2 0 f 0 2
22 2 2
f x 4f x 4 4 4 x f x 2 4 x
2 2
x 0 x 0lim f x 2 lim4 x 0
2 2
f x 2 f x 2 f x 2 2 2
f x 2 2 f x f x 2 2
x 0limf x 2 f 0
1u
x2
2u ux 0
f u1lim x f lim
x u
u 02 2
f u 2 2 f u f u 2 2
2 2
2 2 2
f u 2 2 f u 2 2f u
u u u.
2
2u
f u 2 2lim 0
u
2u
f ulim 0
u.
1u
x2
2u ux 0
f u1lim x f lim 0
x u
3.418. x 2
f x 3g x f x g x x 2 3
x 2
x 2f 2 limf x 3
f f 2 f 2 3
x u
x 2 x 2 u 2 u 2lim f x lim f x lim f u 3 f 2
x u
x 2 x 2 u 2 u 2 u 2
f x 3 f x 3 f u 3 f u 3lim lim lim lim 6
x 2 x 2 u 2 u 2
3.419.
x2
g 32
x
2
lim g x 3
251
1 2 x g x 2 x 2 x 2 3 g x x 1 0
x 0, ,2 2
x2
1lim 3 g x xg x 3
g x 3,
x2
3lim 3 g x xg x 3 3
2
x
2
1lim
g x 3
3.420.
2
x 2 x 2lim x 2 f x lim x x 6 0 4 2 6 3 2
x 2
2 x x 2x 3 2 x 6f x
x 2
3 x 2
x 2
x 2limf x 2 3 5 1 3 2 1
2x 2
x 2 f x x x 6 f x
x 3
x 2 x 3 , x 2 f
x 2 x 2limf x lim x 3 5 f 2 f x x 3 x .
1
uf x
x
1lim 0
f x
2
x u 0
1 u 1lim f x lim
f x u u
3.421. 0
y x : 0 0 0 0 0
f x f x x x x x f x f x x x
0 0 0 0
f x x x f x f x x x
.
0
0x xlim f x f x
1 2
x ,x 1 2
x x 1 1f x x 2 2
f x x ,
1 2 1 2 1 2
f x f x x x x x 1 2
x x 1
252
3.466. 1 x 2 2f x 2x x 3 2 x 3 2
f x x 1
0
x 2 .
:
2
x 2 x 2
lim f x lim 2x x 3 3
2
x 2 x 2
lim f x lim x 1 3 f 2 .
1,3 .
2f 1 2 1 1 3 2 0 , 2
f 3 3 1 8 0 f 1 f 3 0 .
1,3 f 0 .
f x 0 2 32x x 3 0 x 1,2
2 x 1 1,2
2x 1 0 x 1 2,3 x 1 2,3 .
31,3
2.
3.467.
2
x 3 x 3
lim f x lim x x 6 6 f 3
2
x 3 x 3
lim f x lim x 6x 8 1
0,6 2x x 6 0 0,3
x 2 2x 6x 8 0 3,6 x 4 .
3.468.
x 2
lim f x 1,
x 2
lim f x 2 f
x 0,2 f 0 1 3
x 2,4 f 0 3
3.469. 4 3 2f x x 8x x x 3 . f 0 3 , f 1 6 f 0 f 1 0 f
0,1 , f x 0 0,1 .
2xf x x x e x .
f 0 1, f
2 2
f 0 f 0
2 f
0,
2, f x 0
0,
2.
f x 5x 1 2 x f 0 1, f 5 1 f 0 f 0
f 0, , f x 0 0, .
6 4 2 2f x x 8x x x 8 2
f 1 1 0 0 , f 0 8 ,
f 1 f 0 0 f 1,0 , f x 0
1,0 .
3.470. 10 20f x x 2 x 1 x 2 x 1 , x 2,2 .
20f 2 4 2 1 0 , 10
f 2 4 2 1 0 f 2,2
10 20x 1 x 1
f x 0 0x 2 x 2
2,2 .
253
xf x x x e x f 0 1 0 ,
f 0
2 2 f
0,
2,
xx e
f x 0 0x x
0,
2.
3.471.
3 2 2 20 0
0 0 0 0 02 2
0 0
2 x x1 2 x x 1 x 1 x x
1 x x
3 2 2 2f x 2 x 1 x x 1 x x ,
x 0,2
, f 0 2 ,
f 12
,
Bolzano...
3.472. 4 3f x x 3x 2x 5 , x 2,0 .
f 2 16 24 4 5 7 0 f 0 5 0 Bolzano...
3.473. 2 2f x g x 5x f x g x f x g x 5x .
f x g x f x g x 0
5xf x g x
f x g x.
5xh x f x g x
f x g x, x , . ,
5h 0
f g
5h 0
f g
h h 0 . h x 0 f x g x 0
, .
3.474. 2g x f x e 1 lnx x , x 1,e . g 1 f 1 1 2
g e f e e 1 e .
22lnx x f x ln x x 2
2ln1 1 f 1 ln 1 1 0 f 1 1 1 0 g 1 1
2 2 2 22lne e f e ln e e 1 e f e e 1 e e 2e g e 0
Bolzano...
3.475. 3g x f x e x , x 0,1 .
22 f 0 e g 0 0 3 3 3
3 f 1 e 3 e f 1 e 0 g 1 0
g 0 g 1 0 g Bolzano 3g x 0 f x e x
0,1
3.476. f 2 0 , f 1 1, g x f x 3x , x 2,1
g 2 f 2 6 6 0 , g 1 f 1 3 2 0 Bolzano...
3.477. f 1 f 1 0 3 9 3 1 0 0,2
254
3.478.
3 2 3 2
2f x g x x x x x x x
h x x2 2
.
1 2, 2
1 1h
1
3 2 2 2
1 1 1 1 1 1 1f 0 0 .
2
1 1 1h 0 . 2
2 2h
2g 0
3 2
2 2 20 2 2
2 2 2 2
2 2 2h 0 .
1 2
h h 0 h x 0
1 2, .
3.479.
f f x g f x f xe e f f x g f x f x .
f x u f u g u u f x g x x , x .
f x g x x 1 2, .
1 1 1 1f g ,
2 2 2 2f g ,
1 2 1 2
f f 0 . Bolzano...
3.480. 2g x x x 0,1
g 0
g 1 2
g 0 g 1 0
0,1 : g 0
, 0 .
0
2
2
g x x2
.
2 2 2
2
2g 0 g 1 1 1 0
2 2 24
2 20 4 0 4 .
3.481. f f g x f x x .
g f g f f f f f f ,
2
g g f f 0 Bolzano...
3.482. 1 x 2 22 2
f f 2 f f 2 3 2
22
f 2 f 2 12 2
2 f 4 12 f 4 8 0 .
f 2 f 4 0 f 2,4
Bolzano f x 0 2,4 .
1 x 2 2 2f f f f 3
2 2 2 2 2 2 2 2f f 3 f 3 f 2 0 2
f f 0 ,
Bolzano 2
0x , 0
f x 0 .
255
3.483. g 2 g 3 0 Bolzano 0
x 2,3 0g x 0 ,
2
0 0 0 0f x x 5x 6 g x 0 f x 0 x 2,3 .
3.484. x 2f f 0 f 0
x f f 0
x , .
f f 0 f ,f
f 0 , f 0 f f x 0 f x 0 f f x 0 ,
Bolzano f
,x x, , f x 0
, . f 0 f 0 .
3.485. g g g 0 g x 0 x g ,g ,g
g g ,g ,
g g 0 g Bolzano ,
, g 0
3.486. x x1e 4 0 1 xe 4x 0
x.
xf x 1 xe 4x , x 0,2 f 0 1 2
f 2 9 2e 0 f
Bolzano f x 0 x1e 4 0
x
0,2 .
3.487. 2xf x lnx x e
x 0
lim f x 0 1 f 0 .
xlim f x 1 f 0 . f f 0 f
. Bolzano f x 0 , 0,
3.488. x 0 f 3 f 0 0 1
x3
f 1 f 4 0 f 1 f 3 0 f
. Bolzano 1,3 f 0
x3
f 3 f 0
x 13
f 3 f 0
f 3 f 3 0 . Bolzano...
256
3.489. 1
g x f xx
x 0
lim g x
x 0
lim f x f 0 0 1
f 0
xlim g x
xlim f x
x
1lim 0
x
1 f 0 . Bolzano g x 0
, 0, .
3.490.
f x
g xx
, x 0 f x x g x
xlim g x 0 :
x x xlim f x x lim x g x x lim x g x 1 .
x x
,lim f x x lim x g x 1
,
g x f x x
xlim g x 0 : f 0
xlim g x 0 : f 0 g g 0
Bolzano 0 0
x , : g x 0 .
3.491.
2x 2x
2x
2xx x x
f x xe e 1 x 1 1lim lim lim e 1 1
x x xxe
x x
f xlim f x lim x
x
x 0
lim f x 2 , 0,1 f 0 .
xlim f x 1 f 0 Bolzano , .
3.492. 3 2h x f x g x x 3x 3x 6 , x 1,2 .
h 1 1, h 2 4 h
Bolzano 0
x 1,2 : 0 0 0
h x 0 f x g x
3.493. h x f x g x ln x 1 3x 2 , x 0,1 .
h 0 2 , h 1 ln2 1 0 h
0
x 0,1 : 0 0 0
h x 0 f x g x
3.494. x 0
x
f x x 1x
x 0limf x 1 f 0
x 2
2
2 2f 2 6 f 2 32
. 2g x f x x x 2 , x 0,2 .
g 0 f 0 2 1 0 ,
2
g 2 f 2 02
3.495. 2f g f g 1 0 .
257
h x f x g x x x 1 , x , 1 .
, 1
h f g 1 1 g 1 1 g .
1 1 0 . g x 1 x ,
g 1 g 0 h 0 .
h 1 f 1 g 1 1 g 1 1 g 1 1
0 g 1 1 g 1 1 0 h 1 0 .
h h 1 0 . , 1
2h 0 f g 1 0 f g .
3.496. h x f x g x 1, x , . h ,
:
h f g 1 f f 1 1 1 0 h f g 1 f f 1
f f 1 f f 1 f f 1 2 h 2 0 .
h h 0 Bolzano ,
h 0 f g 1 0 f g 1.
3.497. 2g x f x 2x , x , .
2 2 2g f 2 2 , 2 2 2
g f 2 2
g g 0 g 0
x ,
0
g x 0 2
0 0f x 2x .
3.498. 2g x f x x 5x 6 2x 5 , x 2,3 . g 2 1, g 3 1 g
1,2 0
x 2,3
0
g x 0
0
0 2
0 0
5 2xf x
x 5x 6.
3.499. 2h x f x g x x f x g x , x 0,1 .
h 0,1 h 0 f 0 g 0
x 0 10 f 0 11, 12 g 0 13 22 f 0 g 0 24 h 0 0 .
h 1 f 1 g 1 f 1 g 1
x 1 10 f 1 11, 12 g 1 13 22 f 1 g 1 24
120 f 1 g 1 143 f 1 g 1 24 120 f 1 g 1 ,
f 1 g 1 f 1 g 1 0 h 1 0 .
h 0 h 1 0 Bolzano 0,1
h 0 2f g f g .
258
3.500. 2 2 6 2
f x g x 2x f x g x x 2x 1
2 2 6 2f x g x 2xf x 2xg x x 2x 1
2 2 2 2 6f x 2xf x x g x 2xg x x x 1
2 2 6
f x x g x x x 1 1
x 1 1 2 2
f 1 1 g 1 1 0 f 1 1 g 1 1.
x 1 1 2 2
f 1 1 g 1 1 0 f 1 1 g 1 1.
h x 4f x 5g x 1,1
h 1 4f 1 5g 1 4 1 5 1 1 0
h 1 4f 1 5g 1 4 1 5 1 1 0 h 1 h 1 0
Bolzano 1,1 h 0 4f 5g 0 .
3.501. 2g x f x f x x , x 0,2 .
2g 0 f 0 f 0 f 0 f 0 1 0 , 2
g 2 f 2 f 2 2 0 0
3.502.
8 8 8 8e 1 e 1 1 1e 1 1 1 e 1 e ln ln 1 lne
e e e e
1 ln 1 8 1 g 8
h x f x g , x 1,2 h 1 f 1 g 1 g 0
7 1 g 0 h 2 f 2 g 8 g 0 Bolzano...
3.503. 4 2 4x xf x x x 1 1 x
4 2 4 2x x x 1 1 x x x 1 1
f xx x x x
, x 0 .
x 0
1lim f x
2 f
1f 0
2.
0 9f 0 f 1 e 5 f 1
2. g x f x x 5 , x 0,1 .
11
g 0 f 0 5 02
, 1
g 1 f 1 4 02
Bolzano...
3.504. 0
x 0,2 0 0 0 0
f x g x f x g x 0 .
x
h x f x g x f x 12
, 0x 0,2 h 0 f 0 1 h 2 f 2 .
0,2
f 0 ,f 2 .
2xf x 2 x x 2 2
x xf x 2 x x 2 2x 2 x xf x x 2 x .
259
x 0
2 2
x 2 x x 2 xf x
x x x x
2 x 2 xx f x x
x x
x 0
2 xlim x 2
x
x 0
2 xlim x 2
x
x 0
lim f x 2 .
0
x 0
x 0 x 0
lim f x lim f x f 0 2 .
f 0 2 h 0 2 1 3 .
2f x f x 2 x 2 x 0 f 0 f 2 2 f 2 4 .
h 2 4 h 0 h 2 0 0,2
0
x 0,2
0 0 0
h x 0 f x g x .
3.505. h x f x 1 g x x , x , .
,
h f 1 g f g g
g g 1 g 1 1 g .
f x g x x , f g g g 0
0 1 1 0 h 0 .
h f 1 g f 1 f f
f g f 0 0 h 0 .
h h 0 0
x ,
0
h x 0 0 0 0
f x 1 g x x .
3.506. f 0,1
f 0,1 f 0 ,f 1 f 0 0 f 1 1.
g 0,1
g 0,1 g 1 ,g 0 g 0 1 g 1 0 .
2f g g f 2 1 2
f g g f 2 1 0 .
2h x f g x g f x 2x 1, x 0,1 .
h 0,1
h 0 f g 0 g f 0 1 f 1 g 0 1 1 1 1 1 0
h 1 f g 1 g f 1 2 1 f 0 g 1 1 0 0 1 1 0
h 0 h 1 0 , Bolzano 0,1
h 0 2f g g f 2 1
260
3.507. f t , g t
7 t 17 .
x km f 7 0
g 7 x
x km .
f 17 x g 17 0
0
t 7,17 0 0
f t g t .
h t f t g t , t 7,17 . 7,17
h 7 f 7 g 7 x 0
h 17 f 17 g 17 x 0 . h 7 h 17 0 ,
t 7,17 0 0 0 0
h t 0 f t g t 0 f t g t .
3.508.
x 0 x 0
xlim f x lim 1
x,
x 0 x 0
lim f x lim x 1 f 0 1 f
,0 0, f .
g x f x f x f x , x 0, .
g 0 f 0 f f 0 1 f 1 1 1 0 ,
g f f f 1 f 1 1 1 0
3.509. f f 1 0 f f 1 2f f 1 f 1 0 .
f f 1 0 f 0 f 1 0 , 1
f x 0 .
f f 1 0 , 1
f x 0 , 1 .
f x 0 , 1 .
3.510. 2f 0 f 1 f 1 0 .
f 0 f 1 0 f x 0 0,1 .
f 0 f 1 0 f 0 0 f 1 0
3.511. f e f , 2f f e f 0 .
f f 0 f x 0 , .
f f 0 f 0 f 0 .
3.512. 2f f 1 f 0
f f 1 0 f x 0 , 1 .
261
f f 1 0 f 0 f 1 0 1.
3.513. x f x g x h x x , x , .
f g h f g h .
f8
,
g2
,
3
h8
.
3
f g h f g h 0 08 2 8
.
f4
,
g8
,
5
h8
.
5
f g h f g h 0 04 8 8
.
0 .
0 0 0
0
0 f g h 0
0 f g h 0 f g h .
3.514. g x f x x ,
x 0,2
.
g 0 f 0 0 ,
g f 1 1 02
g 0 g 02
.
g 0 g 02
g Bolzano
0x 0,
2
0g x 0 f x x
g 0 g 0 g 0 02
g 02
2.
0x 0,
2
f x x .
3.515. h x f g x g f x 2x , x 0,4 .
h 0 f g 0 g f 0 0 , h 2 f g 2 g f 2 8 f g 2 4 g f 2 4 0
h 0 h 2 0
3.516. g x f x f x 2 , x 1,3 .
g 1 f 1 f 3 , g 3 f 3 f 5 f 3 f 1 f 1 f 3 g 1 g 3 0
3.517. 2 2f x x x ,
2x 0,2 .
262
2
0,2
2f 0 0 , 2 2 2 2 2 2 2
f 2 2 2 1 2 0 .
2f 0 f 2 0 .
2f 0 f 2 0 f 0 0 2
f 2 0 22 f.
2f 0 f 2 0 2 2 2
0,2 : f 0 .
2
0,2 2 2.
3.518.
h x g x g x3
,
x x x
4 2x x x
3 3 3 3 3
2x ,
3.
2h g g g g
3 3
2 2 2 2 2h g g g g
3 3 3 3 3 3
2 2h g g h
3 3
22h h h 0
3.
2h h 0 h 0
3
2h 0
3
2
3.
2h h 0
3
2,
3
2,
3
h 0 .
2, ,
3
h 0
g g 0 g g3 3
.
3.519.
g x f x f x6
,
5x ,
6.
x x5
x7 56x x
6 6 6
2 2g f f
3 3 2,
2g f f
2 2 3,
2g g 0
3 2
263
3.520.
1g x f x f x
1g 0 f 0 f
1 1 2g f f
1 22 2 3g 0 g g ... gg f f
...........................
...........................
1 1g f f 1
1 f 0 f 1 0
1 2g 0 g g ... g 1 0
1g
1 2g g 0
1 2, 0,1 g 0
0,1 g 0
1f f
3.521. h x f x g x , x 0,1 h 0 f 0 g 0 , h 1 f 1 g 1 .
f 0 f 1 g 0 g 1 f 0 g 0 f 1 g 1 h 0 h 1 ,
2h 0 h 1 h 1 0 .
h 0 h 1 0 h 0,1 , 0
x 0,1
0 0 0
h x 0 f x g x .
0
h 0 h 1 0 x 0 0
x 1.
0x 0,1
0 0 0h x 0 f x g x .
3.522. 2 2h x f x 2f x x 3 , x 0,2 .
2h 0 f 0 2f 0 3 f 0 3 f 0 1 0 0 f 0 1.
22
h 2 f 2 2f 2 1 f 2 1 0 . h 0 h 2 0 .
h 0 h 2 0 h 2 0 , h 0 0
h 0 h 2 0 0,2 0,2 :h 0 .
0,2 :h 0 .
3.523. 2g x f x xf x 1 x , x 1,2 2
g 1 f 1 f 1 f 1 f 1 1 0 ,
22
g 2 f 2 2f 2 1 f 2 1 0 g 1 g 2 0
264
3.524.
0 x
0 x20 x
2
.
g x f x f x2
,
x 0,2
.
g 0 f 0 f2
,
g f f f 0 f2 2 2
g 0 g 02
3.525. 0x 2,2
0 0 0 0 0h x x f x g x x 0 .
x f x g x x , x 2,2 . 2,2
2 f 2 g 2 2 2 f 2 g 2 2 .
2 2f x g x 4 x 2,2
2 2f 2 g 2 4 2 2
f 2 g 2 4
2 21 12 f 2 g 2 2 2f 2 g 2 4 2f 2 g 2 f 2 g 2
2 2
21
2 f 2 g 2 02
2 21 12 f 2 g 2 2 2f 2 g 2 4 2f 2 g 2 f 2 g 2
2 2
21
2 f 2 g 2 02
2 2 0 .
A 2 2 0 olzano 0
x 2,2
0 0 0 0
x 0 f x g x x 0 .
2 2 0 2 0 2 0 0
x 2 0
x 2 .
0x 2,2
0x 0
0 0 0f x g x x 0
3.526.
0 0 0 0
x y y x2 2
0 0
y x2
g x f x f x2
,
x ,2
.
g f f2
,
g f f f f2 2
,
g g 0
3.527. f 1,3 . f 1,3 f 1 ,f 3 8,44
1 2
x ,x 4,9 1 2
x x 1 2 1 2
x x x 3 x 3
1 2
1 2 1 2
1 1 5 5g x g x
x 3 x 3 x 3 x 3,
g 4,9 .
x 4
lim g x 5 ,
x 9
lim g x g 4,9 , 5
265
f 0,4
.
2
f 0, f 0 ,f 0,4 4 4
1 2x ,x ,
4 2
1 2x x .
1 2 1 2
x x 2 x 2 x , 1 2 1 2
x x 3 x 3 x ,
1 1 2 2 1 22 x 3 x 2 x 3 x h x h x h ,
4 2
2h , h ,h 3,
4 2 2 4 2
f 1,2 . 1 2x ,x 1,2
1 2
x x 4 4
1 2x x
1 23x 3x 4 4
1 1 2 2 1 2x 3x 1 x 3x 1 f x f x ,
f 1,2 . f 1 5 f 2 23 .
f 1,2 : f 1,2 f 1 ,f 2 5,23 .
f 0,
1 2x ,x 0,
1 2x x :
1 22x 2x
1 2x x ,
1 1 2 2 1 2
2x x 2x x f x f x f 0, .
: f 0 1 f 2 1 f 0, :
f 0, f ,f 0 2 1,1 .
3.528. xf x e x 1821, x f
x
x xlim f x lim e x 1821 ,
x
x xlim f x lim e x 1821 f A .
0 f A xf x 0 e x 1821 0
3.529. f
x
x 1 x 1
lim f x lim e ln x 1 1 ,
x
x xlim f x lim e ln x 1 1 ,
f A .
1 1
xe ln x 1 1 f x 0 f x f 0 x 0
3.530.
1 2x ,x 0,
2
1 2x x
1 2x x ,
1 2 1 2x x x x ,
1 2
f x f x f
0,
2.
f 0 1
f2 2
f 0, ,1
2 2.
266
f 0 1,f 02 2
f 0 f 02
f
f x 0
0,
2f
3.531.
1 2x ,x 0,
2
1 2x x 2 2 2 2
1 2 1 2x x 2x 1 2x 1,
1 2 1 2
x x x x 2 2
1 1 2 22x 1 x 2x 1 x
1 2
g x g x g 0,2
g
0,
2
2
g 0, g 0 ,g 2, 12 2 2
g
g 0,2
g x 0
0,
2.
3.532. 31g x x 1 x
3, x 2, 1
1 2x ,x 2, 1
1 2x x
2 2
1 2 1 2 1 1 2 2
1g x g x x x x x x x 2 0
3 1 2
1 12 x 1 1 x 4 ,
2
2 22 x 1 1 x 4 ,
x1
1 2
2
1 x 21 x x 4
1 x 2 2 2
1 1 2 21 x x x x 2 10
2 2
1 1 2 2x x x x 2 0
1 21 g x g x g
12, 1 .
1
x 2 x 1
1g lim g x , lim g x ,1
3.
1
0 g 1 1
x 1g x 0 .
1,1 1,2
3.533. 2 2f 1 f 4 2f 4 6f 1 10 2 2
f 1 6f 1 f 4 2f 4 10 0
2 22 2
f 1 6f 1 9 f 4 2f 4 1 0 f 1 3 f 4 1 0 .
f 1 3 f 4 1.
1,4 f A f 1 ,f 4 3,1 .
3.534. 2 22 2
f 2 f 5 8f 2 2f 5 17 f 2 4 f 5 1 0 f 2 4 f 5 1.
f f 2 f 5 f
2,5 .
f
2,5 f 2,5 f 5 ,f 2 1,4 .
3.535. f f 4 4 f 8 8 .
267
g g 4 8 g 8 4 .
t x f g h x g f x ,x 4,8
t 4 f g h 4 g f 4 f g 8 g 4 f 4 8 4 8 4 0
t 8 f g h 8 g f 8 f g 4 g 8 f 8 4 8 4 4 0 t 4 t 8 0
268
3.536. f 2,3 , 3,5 , 5,8
f 2 f 3 , f 3 f 5 f 5 f 8 f 2,3 ,
3,5 5,8 .
f 2,3 2,4 , f 3,5 1,4 , f 5,8 1,2 ,
f 2,8 2,4 1,4 1,2 2,4
0 f 2,3 1 1
x 0,2 : f x 0 .
0 f 3,5 1,4 f x 0 3,5
0 f 5,8 1,2 f x 0 5,8 .
f x 0
3.537. f 0,2 , 2,4 , 4,5
f 0 f 2 , f 2 f 4 f 4 f 5
f 0,2 , 2,4
4,5 .
f 0,2 2,1 , f 2,4 2,3 , f 4,5 4,3 ,
f 0,5 2,1 2,3 4,3 4,3
0 f 0,2 1 1
x 0,2 : f x 0 .
0 f 2,4 2,3 2 2
x 2,4 : f x 0
0 f 4,5 4,3 3 3
x 4,5 : f x 0
f x 0 .
3.538. 1,3
f A 3,5 1,3 .
3.539. 3f x x 2x 4 , x 1,2 f 1 1, f 2 8 f
f x 0 1,2 .
f 1,2
4 3f x x 2x 3x 8 x 1,3 f 1 2 , f 3 136 f
f x 0
1,3 . f 1,3
3f x x x 1, x 0, f 0 1, 3
f 1 f
f x 0
0, . f 0,
269
f x x 5x 2 , x 0,1 f 0 2 , f 1 1 3 0 f
f x 0
0,1 . f 0,1
3.540. f x 2lnx 3 x 4 0,1 .
x 0 x 0
lim f x lim 2lnx 3 x 4 0,1 f 0 .
, f 1 2ln1 3 1 4 4 3 1 0
0 12
1 0
,1 0,1 f f 1 0
0
x ,1 0,1 0 0 0
f x 0 2lnx 3 x 4 0 .
1 2
x ,x 0,1 1 2
x x 1 2 1 2
lnx lnx 2lnx 2lnx
1 2 1 2
x x 3 x 3 x 1 2
3 x 4 3 x 4 .
1 1 2 2
2lnx 3 x 4 2lnx 3 x 4 1 2
f x f x f 0,1
3.541. 5g x x 4x 0,1
g 0 0 g 0 5 0 0 5 .
0,1 g 0 .
1 2
x ,x 0,1 1 2
x x 5 5
1 2x x
1 24x 4x , ,
5 5 5 5
1 1 2 2 1 1 2 2 1 2x 4x x 4x x 4x x 4x g x g x .
g 0,1 0,1
3.542. xg x ln e e x x ,1 .
x
xlim e e e 0 e
xlim x
x
x xlim g x lim ln e e x
x
x 1
lim e e 0
x
x 1
lim ln e e ,
x
x 1
lim ln e e x .
1 2
x ,x ,1 1 2
x x , 1 2 1 2 1 2x x x x x xe e e e e e e e
1 2
ln
x xln e e ln e e .
1 2
x x 1 2x x
1 2 1 2ln e e x ln e e x g x g x
,1 .
xx 1
g A lim g x , lim g x ,
, 0 , g x 0
270
3.543.
f
f f f f 0 f2 2
f 1 1
f 2x f x 2x x x2 2 2 2 2
f f
g x f x2
, x ,
f f f f
g f 02 2
,
f f f f
g f 02 2
,
g g 0
3.544. x 1g x f x 2e lnx 3 , x 0, f
xx 0
g A lim g x , lim g x , 0 g A 0
x 0
0x 1
0 0 0g x 0 f x 2e lnx 3 .
3.545.
2 2 2
2 x x x 16f x xf x 4 f x 4 0
2 4 4,
x
g x f x 02
g
3 5
g 3 f 3 02 2
g x 0 x 0 ,
2 2 2 2x x 16 x x 16 x x 16
f x f x f x2 4 2 4 2
, x 0
f 0, .
f 0 2 ,
xlim f x f A , 2 .
k 2 k 2
3.546. f
x 0
lim f x ,
xlim f x f A
f 1 11 x x 1
xlnx 1 x lnx lnx 0 f x f 1 x 1x x
f
1 f 1 f f . 2g x f x f f , x , .
2g f f f f f f 0 ,
2g f f f f f f 0 ,
g g 0 g ,
g 0 2f f f
3.547.
xlim f x
xlim f x f
.
271
3.548. 5f x x 24x 3 .
f 0 3 0 , f 1 20 0 5f 3 3 24 3 3 174 0
0,1 f 0 f 1 0
1 1
0,1 : f 0 1,3
f 1 f 3 0 2 2
1,3 : f 0 .
5x 24x 3 0 0,3 .
3.549. 2 x xf x x e x e x , x , .
2f 1 0, f 0 1 0 , 2
f 1 0 f f 0 0
f 0 f 0 f , f x 0
,0 0, .
3.550. 3 2f x x 3x 1, x 2,1 f 2 3 , f 0 1, f 1 3 0 .
f 2 f 0 0 , f 0 f 1 0 f 2,1 f x 0
2,0 0,1 .
4 2f x 2x x 1, x 2,1 . f 2 35 , f 0 1, f 1 2 .
f 2 f 0 0 , f 0 f 1 0 f 2,1 f x 0
2,0 0,1 .
6 3 2f x x 2x 5x x 1, x 2,1 . f 2 97 , f 0 1, f 1 4 .
f 2 f 0 0 , f 0 f 1 0 f 2,1 f x 0
2,0 0,1 .
3.551. 3 2f x x x 2 , x 1,1 f 1 1 2 0 , f 0 0 0 ,
f 1 3 0 f f x 0
1,0 0,1 .
3.552. 3 2f x x 4x x 3 , x 1,4 f 1 3 , f 0 3 , f 2 3 , f 4 7
f f x 0
1,0 , 0,2 2,4 .
3.553. 3 2f x x x x , f 1 0 , f 0 0 ,
f 1 0 Bolzano 1,0 0,1 .
3.554. 16 16 16 1 0 1 0
3 2f x x x , f 1 1 2 0 , f 0 0 , f 1 1 0
272
3.555. 2g x f x 9 f x 3 f x 3
g 1 5 0 , g 0 7 0 g x 0 1,0 .
h x f x 3 , h 0 1 0 , h 1 7 0 0
x 0,1
0 0
h x 0 f x 3 0 0 0 0
g x f x 3 f x 3 0 g
0,1 .
3.556. 3 2f x 0 x 2x 5x 6 0 x 1 x 3 x 2 .
, 2 2,1 1,3 3,
0x 4 0 2 5
0f x 70 6 4 56
+ +
x , 2f x 0 x x 2 x 1 0 x 1.
2
22 2 2x x 2 x 1 x x 2 x 2x 1 x 1
f 0 2 1 0 f f x 0 x 1,1
f x 0 x 1,1 .
f 2 2 2 3 0 f f x 0 x 1,
f x 0 x 1, .
3.557. f , ,
f f 0 f ,
0
,x : 0f x 0 .
1 0
x x : 2 2 2
0 0 0 0 0 02f x 3f x x x 4 x x 4 0 .
15 0 , f
.
3.558. f 1,2 , 1,2
f f 0 f , . Bolzano ,
f 0 3 2 2 2f 5f f 7 12 7 12 0 3 4
273
3.559. f ,
f f 0 f . Bolzano ,
f 0 2 4 4f 3f e 2 e 2 0
4e 2 0 .
3.560. f 1, , 1,
f f 0 f . Bolzano ,
f 0
2 2f 1 1 1 1
1 1 2
1 2 1
0 1 1.
3.561. . 1 2
x ,x 1 2
x x ,
1 2f x ,f x
1 2x ,x
0 1 2x x ,x
1 2f x f x 0
0 1 2
x x ,x 0f x 0 .
f x h x 2 2 . 0
x x 0 0
f x h x 2 2 0 2
.
3.562. x 1 2 2f 1 1 2 4f 1 3 f 1 4f 1
f 1 f 1 4 0 f 1 0 . ( f x 0 x f 1 4 .
2 4 2 2 4 2f x x 2x 4f x 3 f x 4f x x 2x 3
222 4 2 2
f x 4f x 4 x 2x 1 f x 2 x 1 .
g x f x 2 g x 0
2
2 2g x x 1 0 x
g 1 f 1 2 4 2 0 g x 0 x
2
2 2g x x 1 2 2 2
g x x 1 f x 2 x 1 f x x 3 x .
3.563. 2
2 2x 2 xf x e f x e . 2 2x
f x e 0 f x 0 f
xf x e x
f x e .
f 0 1 0 f x 0 x xf x e , x .
3.564. 2 2 2 2 2 2f x x 2f x f x 2f x x f x 2f x 1 x 1
2 2f x 1 x 1 2
2x 1 0 x , f x 1 0 .
f x 1
2 2 2f x 1 x 1 f x 1 x 1 , x
2 2f x 1 x 1 f x 1 x 1, x .
274
3.565. 22 4 4
f x x 2f x f x 1 x 1 0 f x 1 0 f x 1
f x 1 0 x 4 4f x 1 x 1 f x x 1 1, x
3.566. 2 4 2 4 2f x 4f x x x 4 x x 4 4 x 2 2 4
f x 4f x x 4 x x 4
2 4
f x 2 x x 4 0 f x 2 x
f 0 2 0 2 0 f x 2 x 0 4f x 2 x x 1
4f x 2 x x 1 , x
3.567. 2 x 2 2xf x 2e x e 2xf x 1 2 2 2x x
f x 2xf x x e 2e 1
22 x
f x x e 1 . 22 x
f x x e 1 0 f x x 0
g x f x x g .
xf x x e 1 x
f x x e 1.
g 0 f 0 2 0 g x 0 f x x 0 x .
x xf x x e 1 f x x e 1, x .
3.568. f 0 0 2 f 4 4 f 0
x 0,4
0f x 2 2 2 2 2
0 0 0 0 0 0 0 0f x 4f x x 2x 4 8 x 2x x 2x 4 0
3.569. x 3,3 2 2f x 9 x 1 .
2f x 0 f x 0 1 2
9 x 0 x 3 x 3
f x 0 3,3 3 3 .
3,3
1 2
x ,x 3,3 1 2
x x 1 2f x ,f x
1 2x ,x 3,3
1 2
f x f x 0 1 2
x ,x f 0 .
1 x 2 2 2 2f 9 9 0 9 3 3
1 2
x ,x 3,3 3,3 .
3,3 f x 0
f x 0 x 3,3 f 0 3 f x 0 x 3,3 . 1
2 2 2f x 9 x f x 9 x 2
f x 9 x . f x 0
2f x 9 x x 3,3 .
275
3.570. f f x f x 1 x .
x 0 f 0 f 0 2f 0 0 f 0 0 . x 0
f x 0 f x 0 x ,0 0, f
,0 0, .
f 2 0 f x 0 x 0, . x 2
f 2 f 2 0 f x 0 x ,0 .
2g x 0 f x x 4 0 f x 0 x 0 2 2
x 4 0 x 4 x 2 .
x , 2 0,2 g x 0 ,
x 2,0 2, g x 0 .
2
2 8 4f x x x f x 0 x ,0 f x 0
x 0, 4f x x x 0 4
f x x x 0 .
f 0 0 f
4
4
x , x 0
f x 0 , x 0
x , x 0
.
3.571. 2x 2x 4 16 0 2
x 2x 4 0 x . f
A .
2 2f x 0 x 2x 4 x 0 x 2x 4 x 1
x 0 1
x 0 2
2 2 2 2x 2x 4 x x 2x 4 x x 2
f x 0
f x 0 x , f .
f 1 1 2 4 1 7 1 0 f x 0 x .
3.572. f x 0 f 3 5 0 f x 0 x .
2 2
x x x
f 4 3 x 3x 1 f 4 3 xlim lim lim f 4 3 x
x 5 x.
3.573. 1,4 f x 0 f 0 0 f 3 5
f x 0 x 1,4
3
2x x
f 0 x 2x 5lim lim f 0 x
x 4.
x 2 0 2
f x + +
2x 4 + +
g x + +
276
3.574. f f x 0 x 1,10 f 5 4 0
f x 0 x 1,10 .
f 2 ,f 6 0
4 3 2 4
3 2x x
f 2 x f 8 x 7x 5 f 2 xlim lim
f 6 x x 5x 9 3f 6 x
.
3.575. 1 2
h x f x f x
1 2f x f x h x 0 x h x
.
h x 0 1 2f x f x , x 1 . 1
f x
1 1 2 1 1 1 2 1f f x f f x f f x f f x 2
2 1 1 2f f x f f x
1 1 1 2f f x f f x 3 . 1 2
f x
1 2 2 2 1 2 2 2f f x f f x f f x f f x 4 3 4
1 1 2 2f f x f f x h x 0 .
1 1 2 2f f x f f x x
1 1 2 2f f x f f x
.
3.576. g x f x lnx , x 0,
xlim g x
g 0 .
x 0
lim g x
0,1 g 0 g x 0
,
3.577. 0,2 f 0 0, 0,2
f f 0 0 f f 2 0 . 1
x 0,
2
x ,2 1 2
f x f x 0 1 1.
f x 0 x .
3.578. 0
x : 0f x 0 2 2 2
0 0 0 0 0 0 0f x x f x x x 1 x x 1
2
02x 1
f x 0 f .
2 2 2 2 2 2f x x f x x x 1 f x x x 1 f x 2x 1
2f x 2x 1 , x 2
f x 2x 1, x
3.579. f 0,5 .
f 0 2 f 5 40 2 5 42 f 0 1 f 5
0
x 0,5 : 0f x 1.
277
3.580. f 1 1, f 2 256 , f 1 100 256 f 1,2 ,
0 0
x 1,2 : f x 100
3.581. 2,3 f 2 f 3
f 2 2 f 3 37
3.
0
x 2,3 0
7f x
3.
0
x x 1 2 2
0 0 0 0f x 3f x x 2x 2
0 0
497 x 2x
9
2
0 09x 18x 14 0 ,
3.582. f f 1 3 4 f 2 5 0x 1,2 :
0f x 4
3.583. 2f x 3f x 2 0 f x 1 f x 2 0 f x 1 f x 2
1 2
x ,x 1f x 1 2
f x 2
3
1 22
f 0
x : 0
3f x
2 2
0 0f x 3f x 2 0
9 9
2 0 9 18 8 04 2
f x 1, x f x 2 , x .
3.584. 2
f x 5f x 4 0 f x 1 f x 4 0 f x 1 f x 4 .
1 2
x ,x 1f x 1 2
f x 4
1 3 4 0 1 2
x x ,x 0 2 1
x x ,x 0f x 3
0
x x 2 2
0 0f x 5f x 4 0 3 5 3 4 0 2 0
f x 1, x f x 4 , x .
3.585.
2
x 1 x 1
lim f x lim x 1,
x 1 x 1
lim f x lim 4x 5 1
0
x 1
5
0 5,9
0x 0,3 0
f x 0 .
54x 5 0 x
f x 0 40 x 1
0 x 1
,
2 x 0x 0f x 0
1 x 31 x 3
278
3.586. 1 x 8 1
f 8 f f 8 1 7 f 7 1 f 77
.
1
4 7 f 7 4 f 8 7,8 : f 47
.
1 x f f f 1 1
4f 4 1 f 44
.
3.587. 0,1 g 0 g 1 1 .
g 0 g 1 1 1 1 11
2 2 2 2
g 0 g 1
g 0 g 12
g 0 g 1
g 1 g 02
0,1
g 0 g 1 1 1
g g2 2
.
3.588. 1,8
x 1
f 1 limf x .
2
2x 1 x 1
6x x 16x 6xlim lim
x 1
x 1
3x 1
,
x 1 x 1
6 x 1 6 x 1lim lim
x 1
x 1
3x 1
x 1limf x 3 f 1 2
(2)
f 1 f 8 27 f 8 9
1,8 . 1,8 , f 1 f 8 3 7 9 , f 1 7 f 8 .
0
x 1,8 0f x 7 .
3.589. f f 0 f f 0 , 0 .
0 , ,
f 0 .
,
1x , 1
f x 2x , 2
f x 1 2
x x .
1 2f x f x 0 f olzano
1 2
x ,x , f 0 .
3.590. f m,M
f
1 1f
2 2
1f x
2 x .
3.591. f , m,M :
m f x M x , .
: 1 1
m f x M m f x M
2 2
m f x M m f x M
3 3
m f x M m f x M
279
M :
1 2 3
m f x f x f x M
1 2 3
4m f x f x f x 4M
1 2 3
f x f x f xm M
4.
, :
1 2 3
1 2 3
f x f x f xf f x f x f x 4f
4.
3.592. f , ,
f f x f x , .
: f f f , f f f ,
f f f2
,
f f f2
3f f f f 3f f f2 3
f f f2
3 f f
,
f f f2
f3
.
3.593. f 1,4 m,M : m f x M x 1,4 .
m f 1 M , m f 2 M 3m 3f 2 3M ,
m f 3 M 5m 5f 3 5M , m f 4 M
f 1 3f 2 5f 2 f 4
10m f 1 3f 2 5f 2 f 4 10M m M10
f 1 3f 2 5f 2 f 4
10 f f
0x 1,4
f 1 3f 2 5f 2 f 4f
10.
3.594. f , m,M :
m f x M x , .
: m f M m f M , m f M m f M
f fm m f f M M m M .
f f f f
,
f ff .
280
3.595. f , , m,M :
m f x M x , .
: m f M,
m f M 2m 2f 2M2 2
,
m f M 3m 3f 3M
16m f f f 6M m f f f M
2 6 2
1f f f
6 2 f f
,
1f f f f
6 2.
3.596. f , , m,M :
m f x M x , .
1 2
m f x M, m f x M,..., m f x M
1 2
1 2
f x f x ... f xm f x f x ... f x M m M
1 2f x f x ... f x
f f
,
1 2f x f x ... f x
f .
3.597. f 0,1 , m,M :
m f x M x 0,1 .
1m f M
5,
2m f M
5,
3m f M
5,
4m f M
5
1 2 3 4f f f f
1 2 3 4 5 5 5 54m f f f f 4M m M
5 5 5 5 4
1 2 3 4f f f f
5 5 5 5
4 f f
0,1
1 2 3 4f f f f
5 5 5 5f
4.
3.598. 6,8
1
m f x 2
M f x 1 2x ,x 6,8 .
m f x M x 6,8 .
: 3 3m m m m f 6 f 7 f 8 M M M M 1 .
281
1 30 m f 6 f 7 f 8 M
33f x f 6 f 7 f 8 f x f 6 f 7 f 8 0 1 2x ,x 6,8 .
3.599. f , , m,M :
0 m f x M x , .
1 2
m f x M, m f x M,.....,m f x M
1 2 1 2m f x f x ... f x M m f x f x ... f x M
1 2
f x f x ... f x f f
, 1 2
f f x f x ... f x .
3.600. ,
m,M g x 0 x , m,M 0 .
x , m g x M , 1
x x 1
m g x M , 2
x x
2
m g x M . 2 2
1 2 1 2m g x g x M m g x g x M
1,
1 1 2g g x g x .
1 2
g x g xm M
2
1 2
g x g x
2 1 2
g x ,g x .
1 2
2 2
g x g x, : g
2.
1 2
1 2
g x g xg x g x
2
2
1 2 1 2 1 2 1 2g x g x 2 g x g x g x g x 4g x g x
2 2
1 1 2 2 1 2g x 2g x g x g x 4g x g x
22 2
1 1 2 2 1 2g x 2g x g x g x 0 g x g x 0
3.601. 1 2
x ,x 1 2
f x f x
1 2f x f xe e
1 2f x f x
1 2f x e f x e
1 2 1 2
5 4x 5 4x x x 1 1
3f f x f 5 10x 0
f:1 1
3f f x f 5 10x
3 3f x 5 10x f x 10x 5 0 1 .
1 0,1 . 3g x f x 10x 5 .
0,1 g 0 f 0 5 .
f (0) f(0)f 0 e 5 f 0 5 e f (0) f(0)
g 0 5 e 5 e 0 .
g 1 f 1 10 5 0 g 0 g 1 0 Bolzano
3g x 0 f x 5 10x 0,1 .
282
3.602. x 0
1 x 1
f x xx x
. 1 1 1
x x x x x xx x x
.
x 0 x 0lim x lim x 0 ,
x 0
1lim x 0
x.
x 0
1 xlim 0
x
x 0lim f x 0
1 x 1x , x 0
f x x x
0 , x 0
.
x 0
x 1 1 x 1
x x x x x
x x
1 1lim lim 0
x x
x
xlim 0
x
x x
1
1 x xlim f x lim 0 0 1 11x x
x
x u 0
1
uxlim lim 11 u
x
.
xlim f x 1 0 1, f 0
1 1f 1 0 f
1,
Bolzano f x 0
1, .
3.603. 1 2, 2
x x 0
1 2
1 1 2
1 2
0
2 22 2
1 2 1 2 1 2 1 2 1 2 1 22 2 2
2
1 2 2 1 2 2 1 2
1 2 2 .
0 1 2
f 1 2 2 1,
0 , 1 2
f 1 2 2 1 4
1 4 , 0f
1 , 0.
0 0
lim f lim f 1 f x 0 . f
,0 0,
f .
g x f x lnf x x 2
12,
2.
g 2 f 2 lnf 2 2 2 3 ln3 4 ln3 1 0
1 1 1 1 1 1g f lnf 2 1 ln1 2 0
2 2 2 2 2 2,
1g 2 g 0
2.
283
Bolzano g x 0 f x lnf x 2 x
12,
2.
3.604. 1
h x f x 10 4x3
, x 2,1 .
h 2,1
1
h 2 f 2 10 4 2 f 2 6 03
1
h 1 f 1 10 4 f 1 2 03
,
h 2 h 1 0 Bolzano h x 0
1
f x 10 4x3
2,1 .
MA MB AB g
f 2 6 , x 2 M A
y y
f 1 2 x 1 M B
y y
2 2
MA MB AB 1 1 6 2 5 .
g x MA MB AB 5 g
3.605.
1 2x ,x 0,
2
1 2x x 2 2
1 2x 1 x 1
1 22 x 2 x ,
1 2
f x f x .
2
f 0, f 0 ,f 1, 12 2 4
f 0 1,
2
f 1 02 4
olzano...
f 0 2 f2
0,
2 f 2 .
3.606. 1 u
1 lnx u x e 1 uf u 2u 2e 1 x
f x 2x 2e , x .
f
x 1 f x f 1 0 x 1 f x f 1 0 .
xlim f x ,
xlim f x f A
3.607. 3 2f x f x x 1
0x x 3 2
0 0 0f x f x x 1
3 3 2 2
0 0 0f x f x f x f x x x
2 2 2 2
0 0 0 0f x f x f x f x f x f x 1 x x 1
284
2 2
0 0f x f x f x f x f x
2 2
0 0f x 4f x 0 2 2
0 0f x f x f x f x 0
2 2
0 0f x f x f x f x 1 0 x .
1
2 2
0
0 2 2
0 0
x xf x f x
f x f x f x f x 1
2 2
0 2 2
0 02 2
0 0
x xf x f x x x
f x f x f x f x 1
2 2 2 2
0 0 0x x f x f x x x .
0
2 2
0x xlim x x 0 ,
0
2 2
0x xlim x x 0 ,
0 0
0 0x x x xlim f x f x 0 limf x f x .
,
f f 0 ,
f 0 . x 3 2 2 2f f 1 1 0 1
.
3.608. x 0
235
x
2x 8 x 4 6f x xx x
2 23 3
x 0 x 0
2x 8 x 4 2x 8 2 x 4 2lim lim
x x x
x 0
2 xlim
8 8
x
2
23 3
x 4 4 1
62x 8 2 2 2x 8 4 x x 4 2
,
xu
6
x 0 u 0 u 0
x
u 16lim limx 6u 6
,
.
x 0
1lim f x
6f
1f 0
6.
6xg x f x x
6, x 0,1
1g 0 f 0 0
6,
1
g 1 f 1 1 06
5
x
6f x xx
x 0 . g 0 g 1 0 .
g 0 g 1 0 g 1 0 1.
g 0 g 1 0 g 0,1
g 0
6f( )
6.
0,1 :
6f( )
6.
3.609.
2
x 2 2x x
x
f x 4e f x e e 4 0
e 1,
285
f x 0 f
i. f 0 3 f x 0 2x xf x e e 4 , x .
ii.
x
x 2 2x x x 2
x x x xx x x
e
4f x 4 e e 4 4lim lim lim
3 4 3 4
0
x 2x
x
1 41 16
e e
3
4
016
1
3.610. g
x
x 0 x 0
lim g x lim lnx e x 1 ,
x
x xlim g x lim lnx e x 1 ,
g A
g 0
x A 0,
0g x .
g
g x e g x g 1 x 1
g 1 14 4 2 2 2 2
e e ln 4 ln 4 4 4 g 4 g 4 4 4 2
3.611. ) Bolzano h x g x x 0,1 .
x 1
x
2 2
2
2 2
1 1 1 x 1 1 1 xg x g
x 2 x 22x 2x,
2
x 0
1 1lim x g
x 2.
23
2x 0 x 0
1 3x1x x gx g 3x
x x 7xlim lim
x 2x xx x
x
.
3.612. x , g x , f x ,
f g , g f .
h x f g x 2x g f x , x , .
h f g 2 g f f g g f 0 ,
h f g 2 g f f g g f 0 ,
h h 0 ...
3.613.
22 4 2 2x 2 2x 2 2x
f x x 2x f x e f x x e g x e 0 g x 0
g 0 f 0 1 0 g x 0 2x x x 2g x e e f x e x , x .
286
x x
x 2xx x
e elim lim
e e xe
x
01 e
3.614. f x 0 [3,6] f 3 0 f x 0 x [3,6] .
2g x f x f 3 f 4 , x [3,4] .
g 3 f 3 f 3 f 4 g 4 f 4 f 4 f 3 .
2
g 3 g 4 f 3 f 4 f 3 f 4 0 .
g 3 g 4 0 g 3 0 g 4 0 ,
g 3 g 4 0 olzano...
2h x f x f 5 f 6 .
f 3 f 4 f 5 f 6 2 2
1 2f x f x
1 2f x f x f x 0 ,
1 1.
3.615. 2 2 2
iz w z w iz w iz w z z w w
2i z z izw izw w w z z w w z z izw izw w w z z w w
izw izw 0 zw zw z z
w w
z
w.
zw zw 0 2Im zw 0 Im zw 0 zw f f f f i
f f 0 f f 0 . f ,
Bolzano f x 0 , .
22 2 2 2 2 2 2
z iw z iw z i w i z iw
22 2 2 2 2 2
z w i iz w z w i iz w 2 2 2
z w iz w
3.616. z K 3, 4 2 ,
2 2
3 4 2 .
2 22
f x 3 x 4 x 1, x 0,1 .
f 0 1 0 , 2 2
f 1 3 4 1 2 1 1 0 f 0 f 1 0
f 0,1 2 22
f x 0 3 x 4 x 1
0,1 .
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