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Title: INTRODUCTION TO CALCULUS(LIMITS OF TRIG. FUNCTIONS)
Description: DEFFERENTIATION OF TRIGONOMETRIC FUNCTIONS CALCULUS CHAPTER FOR 1st YEAR BEGGINERS. EVERYTHING REQUIRED
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CHAPTER 04
LIMITS OF TRIGONOMETRIC FUNCTIONS
Students are advised to study Chapter 01 before continuing with this chapter,
Consider a unit circle with an angle θ measured in radians
...
θ = 1
...
1
lim
h 0

sinh
1
h

Proof:
Consider a unit circle with the triangles OAD, OAC and OBC as in the diagram
...
AD
AD AD
= ½
...
sin h
[since sinh =

 AD ]
OA
1
= ½ sinh

Area of circle sector OAC

= ½ r2 h
= ½h

Area of OBC

= ½ base x height
= ½ OC
...
tanh
= ½

[since tanh =

sinh
cosh

From (1) it follows that
½ sinh

<

½h

<

½

49

sinh
cosh

BC BC

 BC ]
OC
1



sinh

<

h <



1

<

h
sinh

sinh
cosh
1
<
cosh

Taking reciprocals give
cos h <

sinh
h

< 1

sinh
 lim 1
h 0
h 0 h
h 0
Using the Pinching Theorem we can conclude that
sinh
lim
1
h 0 h



lim cosh  lim

Theorem 4
...

 0

cos   1
2
cos   1
lim
 0  (cos   1)

lim

 sin 2 
 0  (cos   1)
sin 
sin 
- lim

...
0
0
 (cos   1)



= - lim

 0

Examples
1
...
lim
½ lim
2x0 2x
2x0 2x
½
...
1 = ½

50

cos   1



= -0 = 0

2
...


 sin4x 3x 4x 
lim 

x 0
 4x sin3x 3x 
4
sin4x
3x
lim
lim
3 4x0 4x 3x0 sin3x
4
sin4x
1
lim
4x

0
sin3x
3
4x
lim
3x0 3x
4 1

...

3 1
4
3

Determine limπ[secy  tany]
y 

2

Since the limit is not in the two known forms, we must use trigonometric
identities to transform it into one of the forms
...

2
2
2


We also know that sin( + y) = cosy and cos ( + y) = - siny
2
2
Substituting this in (i), we obtain
1  cos( π2  y)
1  siny 
=
lim
lim 

(  y)0
y 
sin( π2  y)
 cosy 
[1  cos( π2  y)]
1
=
lim
(  y)0 sin( π  y)
1
2

π
- [cos( 2  y) - 1]
2  y
lim
=

(  y)0 sin( π  y)
2
2  y

- [cos( π2  y) - 1]
2  y
lim
lim
=

(  y)0 sin( π  y) (  y)0
2
2  y
[cos( 2  y )  1]
1
=

...
(-1)
...

If y →( -

π
2

π
2

π
2

π
2

π
2

π
2

2

2

51

EXERCISES
Find the following limits if they exist:
sin6x
1
...


lim

3
...


lim

5
...


lim

7
Title: INTRODUCTION TO CALCULUS(LIMITS OF TRIG. FUNCTIONS)
Description: DEFFERENTIATION OF TRIGONOMETRIC FUNCTIONS CALCULUS CHAPTER FOR 1st YEAR BEGGINERS. EVERYTHING REQUIRED
Buy These NotesPreview