1. 1.3Evaluating Limits Analytically

2. Properties of Limits • If f(x) is continuous then direct substitution will find limits • Continuous? • Lines, polynomials, absolute value → yes • Rational, Radical,… → better check • Direct substitution also works for trig functions

3. Constant function like y = 3 All are continuous so these are just direct substitution Try Example 1

4. Other properties: most are obvious… Example 2

5. Example 3

6. Example 4-5

7. General Strategy • Recognize if direct substitution will find the limit at c • If not, try to find an equivalent function where direct substitution will work: • Factor, rationalize, simplify • Use a graph/table to check Problem Set 1.3.1

8. Is there a difference between the graphs of the following functions?

9. Finding Limits: Rationalizing • Factor or eliminate radicals to make an equivalent function • Evaluate this equivalent function for the limit at the undefined value Example Find the limit:

10. Example Find the limit:

11. 2 Useful Trig Limits Example Find the limit: