pythagoras was a greek philosopher who made important developments in mathematics, astronomy, and...
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Pythagoras was a Greek philosopher who made important developments in mathematics, astronomy, and the theory of music. The theorem now known as Pythagoras's theorem was known to the Babylonians 1000 years earlier but he may have been the first to prove it.
Pythagoras of Samos580 – 500 B.C.
Limit Theorems
Suppose that is a constant and the limits
lim ( ) and lim ( )
exist. Thenx a x a
c
f x g x
1. lim ( ) ( ) lim ( ) lim ( )x a x a x a
f x g x f x g x
2. lim ( ) ( ) lim ( ) lim ( )x a x a x a
f x g x f x g x
3. lim ( ) ( ) lim ( ) lim ( )x a x a x a
f x g x f x g x
lim ( )( )4. lim if lim ( ) 0
( ) lim ( )x a
x a x ax a
f xf xg x
g x g x
5. lim ( ) lim ( )x a x a
c f x c f x
6. lim ( ) lim ( ) where is a positive integernn
x a x af x f x n
Limit Theorems
Suppose that is a constant and the limits
lim ( ) and lim ( )
exist. Thenx a x a
c
f x g x
7. limx ac c
8. limx ax a
9. lim , a nonnegative integer.n n
x ax a n
10. lim where is a positive integer
(If is even, we assume that 0.)
n n
x ax a n
n a
11. lim ( ) lim ( ) where is a positive integer
(If is even, we assume that lim ( ) 0.)
n nx a x a
nx a
f x f x n
n f x
Limit Theorems
Suppose that is a constant and the limits
lim ( ) and lim ( )
exist. Thenx a x a
c
f x g x
12. In general, if ( ) is a polynomial or a rational function and
is in the domain of ( ), then
lim ( ) ( ).x a
f x
a f x
f x f a
Example 2
Example 2
1
( )a) lim
( ) ( )t
h x
f x g x
3
1b) lim ( ) 2 ( ) ( )
tf x h x g x
Example 3
2
23
3a) lim
1x
x x
x
2
0
( 5) 25b) lim
h
h
h
2
23
5 6c) lim
9x
x x
x
0
3 3d) lim
x
x
x
2
4 20f) lim
2x
x x
x x
e)
Simplify each limit algebraically using the properties of limits.
g)
Note:
Limit Theorems
13. The Squeeze Theorem
If when is near (except possibly
at ) and
lim lim
then
x a x a
f x g x h x x a
a
f x h x L
lim .x ag x L
These theorems are also true for directional limits.
Example 4
Example 5
Example 6
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