Date post: | 28-Dec-2015 |
Category: |
Documents |
Upload: | derek-rich |
View: | 228 times |
Download: | 1 times |
4
Thank you for not dividing by zero.
• What happens when you "sub in" the value of c in the and the denominator equals zero???
lim ( )x c
f x
2
25
3 10lim
25x
x x
x
For example, this limit.
5
New Techniques to find LimitsNew Techniques to find Limits
2. Rationalizing the numerator
1. Dividing out
3. Special cases
6
Dividing Out Technique: Factor, then reduce.
5
( 5)( 2)lim
( 5)( 5)x
x x
x x
2
25
3 10lim
25x
x x
x
= 7
10
Since we are taking the limit as x approaches 5, and not at x = 5, we do not have to worry about dividing by zero.
Example 1:
7
Dividing Out Technique: Factor, then reduce
0
( 1)( 1)lim
( 1)
x x
xx
e e
e
2
0
1lim
1
x
xx
e
e
2
Since we are again taking the limit as x approaches 0, and not at x = 0, we do not have to worry about dividing by zero.
Direct substitution yields the indeterminate form 0/0.
Factor
Example 2:
8
Rationalizing Technique
0
3 3 3 3lim
3 3x
x x
x x
0
3 3lim
3 3x
x
x x
0
3 3limx
x
x
0
lim3 3x
x
x x
1 3
62 3
We rationalize the numerator instead of the denominator. We are still multiplying by one, thereby not changing the value, just the look.
Example 3:
9
7
2
128lim
2x
x
x
What happens when you substitute x = 2?
Use synthetic to simplify and divide.
6 5 4 3 2
2lim 2 4 8 16 32 64x
x x x x x x
448
Example 8:
10
Transcendental LimitsTranscendental Limits
limsin sin limcsc csc
limcos cos limsec sec
lim tan tan limcot cot
lim limln ln
x c x c
x c x c
x c x c
x x
x c x c
x c x c
x c x c
x c x c
a c x c
11
Special Cases
Theorem 1. 8 The Squeeze TheoremTheorem 1. 8 The Squeeze Theorem
If h(x) < f(x) < g(x) for all x in an open interval containing c, except possible at c itself, and if
lim ( ) lim ( )
then lim ( ) exists and is equal to L.x c x c
x c
h x L g x
f x
19
0
sinlim 1x
x
x
0
1 coslim 0x
x
x
1/
0lim(1 ) x
xx e
The proof is in the book, and uses the squeeze theorem.
You must learn these!
Special limits whose proofs use the squeeze theorem
20
Example 4: 2
0
tanlimx
x
x
Rewrite2
20
sinlim
cosx
x
x x
20
sin sinlim
cosx
x x
x x
20 0
sin sinlim lim
cosx x
x x
x x
0
20
sinlim 1
sin 0and lim
cos 1
x
x
Since
x
x
x
x
= (1)(0) = 0
21
0
(3 3cos )limx
x
x
Direct substitution gives 0/0 which is indeterminate. Rewrite.
0
(1 cos )3lim
x
x
x
0
(1 cos )3lim 3(0) 0
x
x
x
0
(1 cos )lim 0x
xSince
x
Example 5:
22
0
sin5limx
x
x
Multiply the numerator and the denominator by 5.
0
sin5lim
5
5x
x
x 0
s5
ili
n5
5m
x
x
x
= 5(1) = 5
Special case
Example 6: