Benginning Calculus Lecture notes 2 - limits and continuity
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Transcript of Benginning Calculus Lecture notes 2 - limits and continuity
Beginning Calculus- Limits and Continuity -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,FSMT - UPSI
(LECTURE SLIDES SERIES)
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 1 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Learning Outcomes
Determine the existence of limits of functions
Compute the limits of functions
Determine the continuity of functions.
Connect the idea of limits and continuity of functions.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 2 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Limits
Definition 1
The limit of f (x), as x approaches a, equals L, denoted by
limx→a
f (x) = L or f (x)→ L as x → a (1)
if the values of f (x) moves arbitrarily close to L as x moves suffi cientlyclose to a (on either side of a ) but not equal to a.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 3 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→2
(x2 − x + 2
)= 4
0 2 40
5
10
x
y x < 2 f (x) x > 2 f (x)1.0 2.000000 3.0 8.0000001.5 2.750000 2.5 5.7500001.9 3.710000 2.1 4.3100001.99 3.970100 2.01 4.0301001.995 3.985025 2.005 4.0150251.999 3.997001 2.001 4.003001
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 4 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
Estimate the value of limt→0
√t2 + 9− 3t2
.
f
t0.10.0010.00010.00001−0.00001−0.0001−0.001−0.1
=
1t2
(√t2 + 9− 3
)0.166 620.166 670.166 670.166 670.166 670.166 670.166 670.166 62
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 5 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example - continue
4 2 0 2 4
0.12
0.13
0.14
0.15
0.16
limt→0
√t2 + 9− 3t2
=16
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 6 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) = x + 1.
1 1 2 3 4 51
1
2
3
4
5
x
y
limx→2
f (x) = 3
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 7 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
g (x) ={x + 1 if x ≤ 2(x − 2)2 + 3 if x > 2
1 1 2 3 4 51
1
2
3
4
5
x
y
limx→2
g (x) = 3
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 8 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
h (x) ={x + 1 if x < 2(x − 2)2 + 3 if x > 2
1 1 2 3 4 51
1
2
3
4
5
x
y
limx→2
h (x) = 3, eventhough h is not defined at x = 2.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 9 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
One-Sided Limits
Left-hand limit of flimx→a−
f (x) = L (2)
Right-hand limit of flimx→a+
f (x) = L (3)
limx→a
f (x) = L⇔ f limx→a−
f (x) = limx→a+
f (x) = L. (4)
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 10 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) ={x + 1 if x ≤ 2(x − 2)2 + 1 if x > 2
1 1 2 3 4 51
1
2
3
4
5
x
y
limx→2−
f (x) = 3 and limx→2+
f (x) = 1
limx→2
f (x) does not exist (DNE), eventhough f is defined at x = 2.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 11 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
1 1 2 3 4 51
1
2
3
4
5
x
y
Find:
f (2) and f (4)limx→2−
f (x) , limx→2+
f (x) , limx→2
f (x)
limx→4−
f (x) , limx→4+
f (x) limx→4
f (x)
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 12 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Properties of Limits
Suppose that limx→a
f (x) and limx→a
g (x) exists. Then,
1. limx→a
(cf (x)) = c limx→a
f (x) , for any constant c
2. limx→a
[f (x)± g (x)] = limx→a
f (x)± limx→a
g (x)
3. limx→a
[f (x) g (x)] =[limx→a
f (x)] [limx→a
g (x)]
4. limx→a
[f (x)g (x)
]=limx→a
f (x)
limx→a
g (x)provided that lim
x→ag (x) 6= 0
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 13 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Properties of Limits - continue
5. limx→a
x = a
6. limx→a
c = c , for any constant c .
7. limx→a
[f (x)]n =[limx→a
f (x)]nwhere n ∈ Z+.
8. limx→a
n√x = n√a where n ∈ Z+ (If n is even, we assume that a > 0 ).
9. limx→a
n√f (x) = n
√limx→a
f (x) where n ∈ Z+. (If n is even, we assume
that limx→a
f (x) > 0 ).
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 14 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Direct Substitution Property
If f is a polynomial or a rational function and a is in the domain of f ,then
limx→a
f (x) = f (a) (5)
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 15 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→5
(2x2 − 3x + 4
)= lim
x→5
(2x2)− limx→5
3x + limx→5
4
= 2 limx→5
x2 − 3 limx→5
x + limx→5
4
= 2(52)− 3 (5) + 4
= 39
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 16 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→−2
x3 + 2x2 − 15− 3x =
limx→−2
(x3 + 2x2 − 1
)limx→−2
(5− 3x)
=limx→−2
x3 + 2 limx→−2
x2 − limx→−2
1
limx→−2
5− 3 limx→−2
x
=(−2)3 + 2 (−2)2 − 1
5− 3 (−2) = − 111
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 17 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Definition 2
If f (x) = g (x) when x 6= a, then limx→a
f (x) = limx→a
g (x) , provided that
the limits exist.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 18 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→1
x2 − 1x − 1 . For x 6= 1,
x2 − 1x − 1 =
(x − 1) (x + 1)x − 1 = x + 1
limx→1
x2 − 1x − 1 = lim
x→1x + 1 = 2
1 1 2 3 4 51
1
2
3
4
5
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 19 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limh→0
(3+ h)2 − 9h
. For h 6= 0,
(3+ h)2 − 9h
=9+ 6h+ h2 − 9
h= 6+ h
limh→0
(3+ h)2 − 9h
= limh→0
6+ h = 6
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 20 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→2|x − 2|x − 2 .
For x − 2 > 0, |x − 2| = x − 2.
limx→2|x − 2|x − 2 = lim
x→2x − 2x − 2 = lim
x→21 = 1
For x − 2 < 0, |x − 2| = − (x − 2) = 2− x .
limx→2|x − 2|x − 2 = lim
x→2− (x − 2)x − 2 = lim
x→2−1 = −1
limx→2|x − 2|x − 2 DNE
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 21 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Remark 1
limθ→0
cos θ − 1θ
= 0
Rewrite:1− cos θ
θto make the numerator stays positive.
θ1
O
A
BC
BC = 1− cos θ, arclength AB = θ.
1− cos θ
θ→ 0 as θ → 0
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 22 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Remark 2
limθ→0
sin θ
θ= 1
θ1
O
A
BC
AC = sin θ, arclength AB = θ
sin θ
θ→ 1 as θ → 0.
Principle: Short pieces of curves are nearly straight.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 23 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limθ→0
tan θ
θ
tan θ
θ=
sin θ
cos θθ
=sin θ
θ cos θ=sin θ
θ· 1cos θ
limθ→0
tan θ
θ= lim
θ→0sin θ
θ· lim
θ→01cos θ
= 1 · 1 = 1
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 24 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limθ→0
sin 2θ
tan θ
sin 2θ
tan θ=
sin 2θ
θtan θ
θ
=
2 sin 2θ
2θtan θ
θ
limθ→0
sin 2θ
tan θ= lim
θ→0
2 sin 2θ
2θtan θ
θ
=limθ→0
2 sin 2θ
2θ
limθ→0
tan θ
θ
=21= 2
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 25 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Infinite Limits
Definition 3
Let f defined on both sides of a, except possibly at a itself. Then
limx→a
f (x) = ∞ or limx→a
f (x) = −∞ (6)
means that the values of f (x) can be made arbitrarily large (as large aspossible) by taking x suffi ciently close to a, but not equal to a. x = a isthe vertical asymptote.
y
x
y = f(x)
x = aa
y
x
y = f(x)
x = a
a
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 26 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→3+
2xx − 3 = +∞ and lim
x→3−2xx − 3 = −∞
5 5 10
5
5
10
x
y
x = 3
The vertical asymptote is at x = 3.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 27 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) = tan x =sin xcos x
The vertical asymptote can be obtained by setting cos x = 0, that is,
x =π
2x = (2n+ 1)
π
2, n ∈ Z
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 28 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Limits at Infinity
Definition 4 (Limits at Infinity)
(a) Let f be a function defined on some interval (a,∞) . Then
limx→∞
f (x) = L (7)
means that the values of f (x) can be made arbitrarily close to L bytaking x suffi ciently large.
(b) Let f be a function defined on some interval (−∞, a) . Then
limx→−∞
f (x) = L (8)
means that the values of f (x) can be made arbitrarily close to L bytaking x suffi ciently large negative.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 29 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Horizontal Asymptotes
The line y = L is called a horizontal asymptote of the curve y = f (x) ifeither
limx→∞
f (x) = L or limx→−∞
f (x) = L (9)
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 30 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) =x2 − 1x2 + 1
limx→∞
f (x) = 1 = limx→−∞
f (x)
10 5 5 10
1
1
2
x
y
No vertical asymtote.The horizontal asymptote is y = 1.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 31 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) =1x.
limx→0−
1x= −∞, lim
x→0+1x= +∞
limx→∞
1x= 0 = lim
x→−∞
1x
Vertical asymtote at x = 0The horizontal asymptote at y = 0.
4 2 2 4
4
2
2
4
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 32 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example - Finding the Asymptotes
f (x) =3x2 − x − 25x2 + 4x + 1
limx→∞
3x2 − x − 25x2 + 4x + 1
= limx→∞
3x2
x2− xx2− 2x2
5x2
x2+4xx2+1x2
= limx→∞
3− 1x− 2x2
5+4x+1x2
=
limx→∞
(3− 1
x− 2x2
)limx→∞
(5+
4x+1x2
)
=limx→∞
3− limx→∞
1x− limx→∞
2x2
limx→∞
5+ limx→∞
4x+ limx→∞
1x2
=3− 0− 05+ 0+ 0
=35
The horizontal asymptote is y =35.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 33 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example - Finding the Asymptotes
f (x) =
√2x2 + 13x − 5 .
limx→∞
√2x2 + 13x − 5 = lim
x→∞
√2x2 + 1√x2
3x − 5x
,√x2 = x for x > 0
= limx→∞
√2x2
x2+1x2
3xx− 5x
= limx→∞
√2+
1x2
3− 5x
=limx→∞
√2+
1x2
limx→∞
(3− 5
x
) =√limx→∞
2+ limx→∞
1x2
limx→∞
3− limx→∞
5x
=
√23
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 34 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example - Finding the Asymptotes - continue
limx→−∞
√2x2 + 13x − 5 = lim
x→−∞
−√2+
1x2(
3− 5x
) ,√x2 = −x for x < 0
=− limx→∞
√2+
1x2
limx→−∞
(3− 5
x
) = −√23
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 35 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example - Finding the Asymptotes - continue
4 2 2 4
4
2
2
4
x
y
The horizontal asymptotes are: y = ±√23.
The vertical asymptote is when 3x − 5 = 0, that is, x = 53.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 36 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example - Finding the Asymptotes
f (x) =√x2 + 1− x
limx→∞
(√x2 + 1− x
)= lim
x→∞
(√x2 + 1− x
)·
(√x2 + 1+ x
)(√
x2 + 1+ x)
= limx→∞
(x2 + 1
)− x2√
x2 + 1+ x= limx→∞
1√x2 + 1+ x
= limx→∞
1x√
x2 + 1+ x√x2
= limx→∞
1x√
x2
x2+1x2+ 1
= limx→∞
1x√
1+1x2+ 1
=0√
1+ 0+ 1= 0
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 37 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example - Finding the Asymptotes - continue
4 2 0 2 4
5
10
x
y
The horizontal asymptote is y = 0.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 38 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→∞
x3 = ∞ and limx→−∞
x3 = −∞.
4 2 2 4
100
50
50
100
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 39 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→∞
(x2 − x
). Note that the properties of limits cannot be applied to
infinite limits since ∞ is not a number. So,
limx→∞
(x2 − x
)= limx→∞
x (x − 1) = ∞
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 40 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
limx→∞
x2 + x3− x .
limx→∞
x2 + x3− x = lim
x→∞
x2
x+xx
3x− xx
= limx→∞
x + 13x− 1
=∞−1 = −∞
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 41 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Continuous Functions at a Point
Definition 5
A function f is continuous at a if
limx→a
f (x) = f (a) (10)
y
x
y = f(x)
a
f(a)
f (a) is defined (a is in the domain of f )limx→a
f (x) exists.
limx→a
f (x) = f (a)
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 42 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
y
x1 3 50 2 4 6
Discontinuities at 1, 3, and 5.
at a = 1, f is undefined
at a = 3, f is defined but limx→3
f (x) DNE;
at a = 5, f is defined and limx→5
f (x) exists, but limx→5
f (x) 6= f (5) .
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 43 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) =x2 − x − 2x − 2 is discontinuous at 2 because f (2) is undefined.
1 1 2 3 4 51
1
2
3
4
5
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 44 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
g (x) =
{ 1x2
if x 6= 01 if x = 0
is defined at 0 but limx→0
g (x) = limx→0
1x2
does not exist. This discontinuity is called infinite discontinuity.
4 2 2 41
1
2
3
4
5
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 45 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
h (x) =
x2 − x − 2x − 2 if x 6= 2
1 if x = 2is defined at 2 and lim
x→2h (x) = 3,
but limx→2
h (x) 6= h (2) . This discontinuity is called removablediscontinuity.
1 1 2 3 4 51
1
2
3
4
5
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 46 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
k (x) = bxc has discontinuities at all of the integers because limx→n
k (x)
does not exist if n is an integer. These discontinuities are called jumpdiscontinuities.
1 1 2 3 4 51
1
2
3
4
5
x
y
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 47 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Theorem 6
If f and g are continuous at x = a and c is a constant, then thefollowing functions are also continuous at a.
(a) f ± g(b) cf
(c) fg
(d)fgif g (a) 6= 0
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 48 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Theorem 7
The following functions are continuous at every number in their domains.
(a) Polynomial functions.
(b) Rational functions.
(c) Power and root functions
(d) Trigonometric Functions
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 49 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) = x100 − 2x37 + 75 is a polynomial function. So it iscontinuous everywhere: (−∞,∞)
g (x) =x2 + 2x + 17x2 − 1 is a rational function, and continuous on its
domain {x | x 6= ±1} = (−∞,−1) ∪ (−1, 1) ∪ (1,∞) .
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 50 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
h (x) =√x +
x + 1x − 1 −
x + 1x2 + 1
Let h1 (x) =√x ; h2 (x) =
x + 1x − 1 ; and h3 (x) =
x + 1x2 + 1
.
h1 (x) is a root function and continuous on [0,∞).h2 (x) is a rational function and continuous on (−∞, 1) ∪ (1,∞) ,and
h3 (x) is also a rational function and continuous everywhere on R.
So, h (x) is continuous on [0, 1) ∪ (1,∞) .
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 51 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) =sin x
2+ cos x
Let f1 (x) = sin x , and let f2 (x) = 2+ cos x .
f1 (x) and f2 (x) are trigonometric functions. So, they arecontinuous. Note that cos x ≥ −1. So, f2 (x) = 2 cos x is alwayspositive.
Hence, f (x) =f1 (x)f2 (x)
=sin x
2+ cos xis continuous everywhere on R.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 52 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Theorem 8
If g is continuous at a and f is continuous at g (a) , then(f ◦ g) (x) = f (g (x)) is continuous at a.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 53 / 54
The Limit of a Function Limits of Trigonometric Functions Infinite Limits and Limits at Infinity Continuity
Example
f (x) = sin(x2)
Let F (x) = sin x , and let G (x) = x2.
F and G are continuous on R.
So, f (x) = F (G (x)) = sin(x2)is continuous on R.
VillaRINO DoMath, FSMT-UPSI
(D1) Limits and Continuity 54 / 54