1. Limits Review

2. Limit L a

3. c) Find 6
Note: f (-2) = 1
is not involved 2

4. One-Sided Limit One-Sided Limits
The right-hand limit of f (x), as x approaches a, equals L
written:
if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. L a

5. The left-hand limit of f (x), as x approaches a, equals M
written:
if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. M a

6. Examples of One-Sided Limit
1. Given
Find
Find

7. More Examples
Find the limits:

8. A Theorem This theorem is used to show a limit does not exist.
For the function

9. Indeterminate Forms
Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions.
Notice form
Ex.
Factor and cancel common factors

10. More Examples

11. Evaluate the limits
Answer : 16
Answer : DNE
Answer :∞
Answer : 1/2

12. Limits at Infinity Ex. Degree on top larger: infinity
Degree on bottom bigger: 0
Degrees equal: Look at lead coefficients
Ex. =2 =0

14. Infinite Limits The limit from the left is -∞
The limit from the right is ∞
The limit doesn’t exist
The limit from the left is ∞
The limit from the right is ∞
The limit = ∞

15. Plug in a number to the left of 3
Examples
Find the limits
Plug in a number to the left of 3