Finding Limits Analytically

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. • Example:

4. Basic Limits: 1. 2. 3. 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: 1. 2. Examples:

16. Steps to follow for finding Limits: