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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

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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

7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions

10/106

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

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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

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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

Example

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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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47/106

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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48/106

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

7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions

52/106

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

7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions

56/106

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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57/106

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

7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions

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

7/27/2019 M53 Lec1.5 Limits and Continuity of Trigonometric Functions

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