1. Finding Limits Analytically
1.3

2. Concepts Covered: Properties of Limits Strategies for finding limits
The Squeeze Theorem

3. From section 1.2: The limit of f(x) as x approaches c does not depend on the value of f at x = c.
Example:
However, sometimes the limit may be exactly f(c). When this occurs the function is at x = c. In this case find the limit by substitution.

4. Basic Limits:
Examples:

5. Properties of Limits: Scalar Multiple: Sum or Difference: Product:
Quotient:
Power:

6. Example using limit properties:

7. Example using limit properties:
Find the following:

8. Example using limit properties:

9. Limits of Trigonometric Functions:
You can evaluate the limits of trig functions using direct substitution if
Examples:

10. Did you understand?
When can you use direct substitution to find a limit?

11. If direct substitution will not work….
Try one of these techniques:
Cancellation
Rationalization

12. Cancellation:
Try to find a function that agrees with your function at all but .
Example:

13. Rationalization:
Rationalize the numerator.

14. The Squeeze Theorem:
If h(x) ≤ f(x) ≤ g(x) for all x in an open interval containing c, except for possibly at c itself and if , then
Example: If 4 – x2 ≤ f(x) ≤ 4 + x2, find

15. Two Special Trig Limits:
Examples:

16. Steps to follow for finding Limits: