Download Special Trigonometric Limits

CALCULUS
CHAPTER 2: 2.1 Limits (Computational Techniques – Day 4)


Establish the limits of some simple functions.
Develop some theorems that will enable us to use the limits of simple functions as building blocks for
finding limits of more complicated functions.
Trigonometric Functions: If
is any real number in the domain of the given function, then
Example: Compute the limit of
at
We have
Special Trigonometric Limits:
This topic states the limits of the trigonometric functions and also two very useful limits involving sine and cosine.
These special limits can either be proved with L'Hopital’s rule or with the squeeze rule. Examples are given which
illustrate there usefulness.
The following trigonometric limits hold:
Example: Find the limits.
sin 2 x
x 0
2x
lim
sin x 1  cos x 
x 0
2 x2
lim
sin x 2 1  cos x 
x 0
2x2
lim
sin 5 x
x 0 sin 4 x
lim
sin x
1
x 0
x
Know this… lim
.
Evaluate each limit.
sin x
2x
sin x
x
2
2. lim
5. lim
sin 3 x
x 0
3x
6. lim
9. lim 1  cos 4 x 
10. lim
1.
lim
x 
x 0
x 0
sin 3 x
x 0
2x
x 0

sin x

 sin x
12. f  x   
 x
 1
cos x

3. lim
x 0
x
41  cos x 
x
4. lim
x 0
sin 2 x
x  0 sin 3 x
sin 5 x
x 0
x
7. lim
tan x
x
11. lim
x 0
cos x tan x
x
8. lim
cos 2 x
2

6

 x 
6
x 
a) lim f  x 
x

b) lim f  x 
x 
6
c) lim f  x 
x

d) lim f  x 
x  2
4
x 
e)
f  
 

6
f) f 
Assignment: page 66, 67-77 odd