benginning calculus lecture notes 2 - limits and continuity
DESCRIPTION
Universiti Pendidikan Sultan IdrisTRANSCRIPT
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Beginning Calculus- Limits and Continuity -
Shahrizal Shamsuddin Norashiqin Mohd Idrus
Department of Mathematics,FSMT - UPSI
(LECTURE SLIDES SERIES)
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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.
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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.
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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
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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
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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
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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
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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
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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.
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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)
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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.
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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)
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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
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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 ).
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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)
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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
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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
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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.
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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
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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
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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
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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
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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.
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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
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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
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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
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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.
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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
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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.
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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)
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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.
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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
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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.
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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
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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
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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.
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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
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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.
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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
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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) = ∞
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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 = −∞
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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)
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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) .
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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
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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
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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
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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
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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
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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
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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,∞) .
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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,∞) .
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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.
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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.
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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.
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