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Limits and Continuity of Trigonometric Functions
Mathematics 53
Institute of Mathematics (UP Diliman)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 1 / 19
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For today
1 Limits and Continuity of Trigonometric Functions
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 2 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1.
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1. Item 1 is used to prove item 2
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1. Item 1 is used to prove item 2,
and item 2 is used to prove item 3.
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1. Item 1 is used to prove item 2,
and item 2 is used to prove item 3.Finally,
limx0
1 cos x
x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1. Item 1 is used to prove item 2,
and item 2 is used to prove item 3.Finally,
limx0
1 cos x
x= lim
x0
1 cos x
x1 + cos x
1 + cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1. Item 1 is used to prove item 2,
and item 2 is used to prove item 3.Finally,
limx0
1 cos x
x= lim
x0
1 cos x
x1 + cos x
1 + cos x
= lim
x0
1 cos2 x
x(1 + cos x)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1. Item 1 is used to prove item 2,
and item 2 is used to prove item 3.Finally,
limx0
1 cos x
x= lim
x0
1 cos x
x1 + cos x
1 + cos x
= lim
x0
1 cos2 x
x(1 + cos x)
= limx0
sin xx
sin x
1 + cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1. Item 1 is used to prove item 2,
and item 2 is used to prove item 3.Finally,
limx0
1 cos x
x= lim
x0
1 cos x
x1 + cos x
1 + cos x
= lim
x0
1 cos2 x
x(1 + cos x)
= limx0
sin xx
sin x
1 + cos x
= limx0
sin x
x limx0
sin x
1 + cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
limx0
sin x
x= 1
limx0 sin x = 0
limx0
cos x = 1
limx0
1 cos xx
= 0
The Squeeze Theorem is used to prove the item 1. Item 1 is used to prove item 2,
and item 2 is used to prove item 3.Finally,
limx0
1 cos x
x= lim
x0
1 cos x
x1 + cos x
1 + cos x
= lim
x0
1 cos2 x
x(1 + cos x)
= limx0
sin xx
sin x
1 + cos x
= limx0
sin x
x limx0
sin x
1 + cos x
= 1
0
2
= 0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 3 / 19
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Theorem
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 4 / 19
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Theorem
1 limxa
cos x = cos a
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 4 / 19
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Theorem
1 limxa
cos x = cos a and limxa
sin x = sin a
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 4 / 19
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Theorem
1 limxa
cos x = cos a and limxa
sin x = sin a
2 The trigonometric functions are continuous on their respective domains.
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 4 / 19
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Example
Evaluate: limx0
sin (4x)
x
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 5 / 19
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Example
Evaluate: limx0
sin (4x)
x
Solution:
limx0
sin (4x)
x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 5 / 19
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Example
Evaluate: limx0
sin (4x)
x
Solution:
limx0
sin (4x)
x= lim
x0
sin (4x)
x4
4
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 5 / 19
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Example
Evaluate: limx0
sin (4x)
x
Solution:
limx0
sin (4x)
x= lim
x0
sin (4x)
x4
4
= lim
x0
sin (4x)
4x (4)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 5 / 19
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Example
Evaluate: limx0
sin (4x)
x
Solution:
limx0
sin (4x)
x= lim
x0
sin (4x)
x4
4
= lim
x0
sin (4x)
4x (4) = 1(4)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 5 / 19
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Example
Evaluate: limx0
sin (4x)
x
Solution:
limx0
sin (4x)
x= lim
x0
sin (4x)
x4
4
= lim
x0
sin (4x)
4x (4) = 1(4) = 4
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 5 / 19
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Evaluate: limx0
x
sin x
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 6 / 19
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Evaluate: limx0
x
sin x
Solution:
limx0
xsin x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 6 / 19
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Evaluate: limx0
x
sin x
Solution:
limx0
xsin x
= limx0
1
sin xx
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 6 / 19
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Evaluate: limx0
x
sin x
Solution:
limx0
xsin x
= limx0
1
sin xx
= 1
1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 6 / 19
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Evaluate: limx0
x
sin x
Solution:
limx0
xsin x
= limx0
1
sin xx
= 1
1= 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 6 / 19
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RemarkIn general, if lim
xaf(x) = 0 and f(x) = 0 for any x in some open interval
containing a (except possibly at a), then
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 7 / 19
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RemarkIn general, if lim
xaf(x) = 0 and f(x) = 0 for any x in some open interval
containing a (except possibly at a), then
limxa
sin[f(x)]
f(x) = 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 7 / 19
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RemarkIn general, if lim
xaf(x) = 0 and f(x) = 0 for any x in some open interval
containing a (except possibly at a), then
limxa
sin[f(x)]
f(x) = 1
limxa
f(x)
sin[f(x)]= 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 7 / 19
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RemarkIn general, if lim
xaf(x) = 0 and f(x) = 0 for any x in some open interval
containing a (except possibly at a), then
limxa
sin[f(x)]
f(x) = 1
limxa
f(x)
sin[f(x)]= 1
limxa
1 cos[f(x)]
f(x) = 0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 7 / 19
Example
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Example
Evaluate: limx0
sin(3x)
sin(5x)
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 8 / 19
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Example
Evaluate: limx0
sin(3x)
sin(5x)
Solution:
limx0
sin(3x)
sin(5x)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 8 / 19
Example
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Example
Evaluate: limx0
sin(3x)
sin(5x)
Solution:
limx0
sin(3x)
sin(5x)3x
3x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 8 / 19
Example
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Example
Evaluate: limx0
sin(3x)
sin(5x)
Solution:
limx0
sin(3x)
sin(5x)3x
3x5x
5x= lim
x0
sin(3x)
3x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 8 / 19
Example
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Example
Evaluate: limx0
sin(3x)
sin(5x)
Solution:
limx0
sin(3x)
sin(5x)3x
3x5x
5x= lim
x0
sin(3x)
3x
5x
sin(5x)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 8 / 19
Example
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p
Evaluate: limx0
sin(3x)
sin(5x)
Solution:
limx0
sin(3x)
sin(5x)3x
3x5x
5x= lim
x0
sin(3x)
3x
5x
sin(5x)3
5
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 8 / 19
Example
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p
Evaluate: limx0
sin(3x)
sin(5x)
Solution:
limx0
sin(3x)
sin(5x)3x
3x5x
5x= lim
x0
sin(3x)
3x
5x
sin(5x)3
5= (1)(1)
3
5
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 8 / 19
Example
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p
Evaluate: limx0
sin(3x)
sin(5x)
Solution:
limx0
sin(3x)
sin(5x)3x
3x5x
5x= lim
x0
sin(3x)
3x
5x
sin(5x)3
5= (1)(1)
3
5
=
3
5
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 8 / 19
Example
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p
Evaluate: limx0
tan x
x
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 9 / 19
Example
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Evaluate: limx0
tan x
x
Solution:
limx0
tan x
x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 9 / 19
Example
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Evaluate: limx0
tan x
x
Solution:
limx0
tan x
x= lim
x0
sin x
x
1
cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 9 / 19
Example
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Evaluate: limx0
tan x
x
Solution:
limx0
tan x
x= lim
x0
sin x
x
1
cos x
= 1(1)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 9 / 19
Example
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Evaluate: limx0
tan x
x
Solution:
limx0
tan x
x= lim
x0
sin x
x
1
cos x
= 1(1) = 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 9 / 19
Example
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Evaluate: limx0
x2
sin (3x2)
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 10 / 19
Example
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Evaluate: limx0
x2
sin (3x2)
Solution:
limx0
x2
sin (3x2)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 10 / 19
Example
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Evaluate: limx0
x2
sin (3x2)
Solution:
limx0
x2
sin (3x2)= lim
x0
x2
sin (3x2)3
3
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 10 / 19
Example
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Evaluate: limx0
x2
sin (3x2)
Solution:
limx0
x2
sin (3x2)= lim
x0
x2
sin (3x2)3
3
= lim
x0
3x2
sin (3x2)1
3
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 10 / 19
Example
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Evaluate: limx0
x2
sin (3x2)
Solution:
limx0
x2
sin (3x2)= lim
x0
x2
sin (3x2)3
3
= lim
x0
3x2
sin (3x2)1
3=
1
3
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 10 / 19
Example
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Evaluate: limx1
sinx2 1
x + 1
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 11 / 19
Example
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Evaluate: limx1
sinx2 1
x + 1
Solution:
limx1
sinx2 1
x + 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 11 / 19
Example
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Evaluate: limx1
sinx2 1
x + 1
Solution:
limx1
sinx2 1
x + 1
= limx1
sin
x2 1
x + 1
x 1
x 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 11 / 19
Example
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Evaluate: limx1
sinx2 1
x + 1
Solution:
limx1
sinx2 1
x + 1
= limx1
sin
x2 1
x + 1
x 1
x 1
= limx1
sin
x2 1
x2 1 (x 1)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 11 / 19
Example
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Evaluate: limx1
sinx2 1
x + 1
Solution:
limx1
sinx2 1
x + 1
= limx1
sin
x2 1
x + 1
x 1
x 1
= limx1
sin
x2 1
x2 1 (x 1) = 1(2)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 11 / 19
Example
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Evaluate: limx1
sinx2 1
x + 1
Solution:
limx1
sinx2 1
x + 1
= limx1
sin
x2 1
x + 1
x 1
x 1
= limx1
sin
x2 1
x2 1 (x 1) = 1(2) = 2
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 11 / 19
Example
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7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
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Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
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Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
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58/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
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59/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
60/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
61/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
62/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x2x
2x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
63/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x2x
2x
= lim
x0+
sin(3x)
3x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
64/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x2x
2x
= lim
x0+
sin(3x)
3x
2x
sin(2x
)2
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
65/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x2x
2x
= limx0+
sin(3x)
3x
2x
sin(2x)2
1
x cos(3x)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
66/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) = limx0+tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x2x
2x
= limx0+
sin(3x)
3x
2x
sin(2x)2
1
x cos(3x)3
4
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
67/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) =lim
x0+
tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x2x
2x
= limx0+
sin(3x)
3x
2x
sin(2x)2
1
x cos(3x)3
4
=
(1)(1)2
1
(0 )(1)
3
4
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
68/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) =lim
x0+
tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x2x
2x
= limx0+
sin(3x)
3x
2x
sin(2x)2
1
x cos(3x)3
4
=
(1)(1)2
1
(0+)(1)
3
4
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
69/106
Evaluate: limx0+
tan(3x)1 cos2(2x)
Solution:
limx0+
tan(3x)
1 cos2(2x) =lim
x0+
tan(3x)
sin2(2x)
= limx0+
sin(3x)
cos(3x) sin2(2x)3x
3x2x
2x2x
2x
= limx0+
sin(3x)
3x
2x
sin(2x)2
1
x cos(3x)3
4
= +
(1)(1)2
1
(0+)(1)
3
4
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 12 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
70/106
Evaluate: limx0
1 cos x
x2
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 13 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
71/106
Evaluate: limx0
1 cos x
x2
Note:
limx0
1 cos x
x2= lim
x0
1cos xx
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 13 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
72/106
Evaluate: limx0
1 cos x
x2
Note:
limx0
1 cos x
x2= lim
x0
1cos xx
x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 13 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
73/106
Evaluate: limx0
1 cos x
x2
Note:
limx0
1 cos x
x2= lim
x0
1cos xx
x
0
0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 13 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
74/106
Evaluate: limx0
1 cos x
x2
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 14 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
75/106
Evaluate: limx0
1 cos x
x2
Solution:
limx0
1 cos x
x2= lim
x0
1 cos x
x21 + cos x
1 + cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 14 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
76/106
Evaluate: limx0
1 cos x
x2
Solution:
limx0
1 cos x
x2= lim
x0
1 cos x
x21 + cos x
1 + cos x
= limx0
1 cos2 xx2
11 + cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 14 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
77/106
Evaluate: limx0
1 cos x
x2
Solution:
limx0
1 cos x
x2= lim
x0
1 cos x
x21 + cos x
1 + cos x
= limx0
1 cos2 xx2
11 + cos x
= limx0
sin2 x
x2
1
1 + cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 14 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
78/106
Evaluate: limx0
1 cos x
x2
Solution:
limx0
1 cos x
x2= lim
x0
1 cos x
x21 + cos x
1 + cos x
= limx0
1 cos2 xx2
11 + cos x
= limx0
sin2 x
x2
1
1 + cos x
= limx0 sin
xx2
1
1 + cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 14 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
79/106
Evaluate: limx0
1 cos x
x2
Solution:
limx0
1 cos x
x2= lim
x0
1 cos x
x21 + cos x
1 + cos x
= limx0
1 cos2 xx2
11 + cos x
= limx0
sin2 x
x2
1
1 + cos x
= limx0 sin
xx2
1
1 + cos x
= (1)21
2
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 14 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
80/106
Evaluate: limx0
1 cos x
x2
Solution:
limx0
1 cos x
x2= lim
x0
1 cos x
x21 + cos x
1 + cos x
= limx0
1 cos2
xx2
11 + cos x
= limx0
sin2 x
x2
1
1 + cos x
= limx0 sin
xx2
1
1 + cos x
= (1)21
2
=
1
2
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 14 / 19
Examples
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
81/106
limx
4
tan x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 15 / 19
Examples
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
82/106
limx
4
tan x = tan
4
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 15 / 19
Examples
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
83/106
limx
4
tan x = tan
4
= 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 15 / 19
Examples
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
84/106
limx
4
tan x = tan
4
= 1
limx
1 cos x
x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 15 / 19
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
85/106
Examples
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
86/106
limx
4
tan x = tan
4
= 1
limx
1 cos x
x =
1 cos
=
2
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 15 / 19
Examples
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
87/106
limx
4
tan x = tan
4
= 1
limx
1 cos x
x =
1 cos
=
2
limx2
sin
x2 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 15 / 19
Examples
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
88/106
limx
4
tan x = tan
4
= 1
limx
1 cos x
x =
1 cos
=
2
limx2
sin
x2 1
= sin(4 1)
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 15 / 19
Examples
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
89/106
limx
4
tan x = tan
4
= 1
limx
1 cos x
x =
1 cos
=
2
limx2
sin
x2 1
= sin(4 1) = sin 3
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 15 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
90/106
Evaluate limx+
cos
1
x
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 16 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
91/106
Evaluate limx+
cos
1
x
Solution:
limx+
1
x = 0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 16 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
92/106
Evaluate limx+
cos
1
x
Solution:
limx+
1
x = 00 is in the domain of cos x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 16 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
93/106
Evaluate limx+
cos
1
x
Solution:
limx+
1
x = 00 is in the domain of cos x
Therefore,
limx+
cos1
x = cos(0) = 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 16 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
94/106
Evaluate limx0
tan
1 cos x
x
Solution:
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 17 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
95/106
Evaluate limx0
tan
1 cos x
x
Solution:
limx0
1 cos x
x = 0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 17 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
96/106
Evaluate limx0
tan
1 cos x
x
Solution:
limx0
1 cos x
x =0
0 is in the domain of tan x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 17 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
97/106
Evaluate limx0
tan
1 cos x
x
Solution:
limx0
1 cos x
x =0
0 is in the domain of tan x
Therefore,
limx0
tan1 cos x
x = tan(0) = 0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 17 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
98/106
Evaluate limx+
cot x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 18 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
99/106
Evaluate limx+
cot x
Solution:
Note that / domcot.
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 18 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
100/106
Evaluate limx+
cot x
Solution:
Note that / domcot. But,
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 18 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
101/106
Evaluate limx+
cot x
Solution:
Note that / domcot. But,
limx+
cot x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 18 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
102/106
Evaluate limx+
cot x
Solution:
Note that / domcot. But,
limx+
cot x = limx+
cos x
sin x
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 18 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
103/106
Evaluate limx+
cot x
Solution:
Note that / domcot. But,
limx+
cot x = limx+
cos x
sin x
1
0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 18 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
104/106
Evaluate limx+
cot x
Solution:
Note that / domcot. But,
limx+
cot x = limx+
cos x
sin x
1
0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 18 / 19
Example
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
105/106
Evaluate limx+
cot x
Solution:
Note that / domcot. But,
limx+
cot x = limx+
cos x
sin x= +
1
0
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 18 / 19
Exercises
-
7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions
106/106
Evaluate the following limits:
1 limx0
4x
sin(5x)
2 limx
sin x2
5x
3 limx0
sin3(3x)
x2 tan(2x)
4 limx2
sin(2 x)
x2 x 2
5 limx0
2x sin x1 cos(4x)
6 limx1
sin x sin 1
x 1
Institute of Mathematics (UP Diliman) Trigo Functions: Limits and Continuity Mathematics 53 19 / 19