MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag Example: 11 0.110 0.01100 0.0011000...
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MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag Example : x 1 lim : Find x 1 1 10 0.1 100 0.01 1000 0.001 ----- ------ 1,000,0 00 0.00000 1 x x 1 0 1 lim x x Example : 2 x 1 lim : Find x 1 1 10 0.01 100 0.0001 1000 0.00000 1 ----- ------ 1,000,0 00 10^(- x 2 1 x 0 1 lim 2 x x 0 1 lim 0 if : Theorem x r x r
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Transcript of MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag Example: 11 0.110 0.01100 0.0011000...
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
Example:
x
1lim :Findx
11
100.1
1000.01
10000.001
-----------
1,000,0000.000001
xx
1
01
limx
x
Example:
2x
1lim :Find
x
11
100.01
1000.0001
10000.000001
-----------
1,000,00010^(-12)
x 21x
01
lim2x
x
01
lim
0 if :Theorem
x
rx
r
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
defined is s.t
01
lim
0 if :Theorem
x
r
r
x
x
r
01
lim
0 if :Theorem
x
rx
r
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
Example:
2
xlim :Find x
Example:
3
xlim :Find x
Example:
2
xlim :Find x
Example:
3
xlim :Find x
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
Example:
1
1lim :Find
2
2
x
x
xTo evaluate the limit at infinity of any rational function, we first divide both thenumerator and denominator by the highest power of that occurs in the denominator.
Example:
145
23lim :Find
2
2
x
xx
xx
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
To evaluate the limit at infinity of any rational function, we first divide both thenumerator and denominator by the highest power of that occurs in the denominator.
Example:
53
12lim :Find
2
x
x
x
Example:
12
43lim :Find
2x
x
xExample:
53
12lim :Find
2
x
x
x
xxx 2 ,x
xxx 2 ,x
Notes: )()(
)()(
)()( )()( )()(
)(c
0
0
0
c
cund
c
)(c c
Notes:
)()(
)()(
)(0
undund
und und
Notes:
Notes:
2
1
0
e
0 und
0 und
e 01
e
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
Multiply by conjugate radical.Example:
xx
1lim :Find 2
x
Example:
x
xx
3lim :Find
2
x
Example:
)(lim :Find 2
xxx
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
Example:
xe1
0xlim :Find
Example:
xsinlim :Findx
Example:
xex
lim :Find
The line
Is a horizontal asymptote
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
01
limx
x
01
limx
x 0y
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
The line
Is a horizontal asymptote
1yExample:
1
1lim :Find
2
2
x
x
x
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
The line
Is a horizontal asymptote5
3y
Example:
145
23lim :Find
2
2
x
xx
xx
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
Example:
53
12lim :Find
2
x
x
x
3
2 ,
3
2 yy
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
To evaluate the limit at infinity of any rational function, we first divide both thenumerator and denominator by the highest power of that occurs in the denominator.
Multiply by conjugate radical.
Factor then take the limit
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
Who is going faster to infinity
Example:
xy
2xy
3xy
xey
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
Example: Sketch the graph of
234 )4()2()1( xxxxy
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
R
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
HH
R
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
H
H
H
H
H
R
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
H
H
H
R
R
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
MATH 101- term 101 : CALCULUS I – Dr. Faisal Fairag
