Chapter 2 Section 3
Suppose that:
g(x) £
f(x)
£
h(x)
lim g(x)
= lim h(x) = L for all x
¹
c
x®c
x®c
then lim f(x) = L
x®c
RULE:
| lim | sin q | = 1 |
| q®0 | q |
Examples:
| 1. lim | sin 3x | ®3 | sin 3x | =3 | lim | sin 3x | = | 3·1 = 3 |
| x®0 |
x |
3x |
x®0 |
3x |
| 2. lim | tan 5x | ® | sin5x | · | cos2x | = | sin5x . 1 | · | cos2x |
| x®0 | tan 2x | cos5x | sin2x |
sin2x |
cos5x |
|
® |
sin 5x | · | 2x | · | 5x | · | cos2x |
| 5x | sin2x | 2x | cos5x |
|
® |
1 |
· | 1 | · | 5 | · | 1 = | 5 |
| 2 | 2 |
| 3. lim | x2 + 1 | ® | 02 + 1 | = | 1 | = | 1 |
| x®0 |
1-sin x |
1-sin 0 |
1 |
 |
 |