1. In previous sections we have been using calculators and graphs to guess the values of limits. Sometimes, these methods do not work! In this section we will use properties of limits, called limit laws, to calculate limits.

2. Limit Laws: Suppose that c is a constant and the limits and exist. Then 1. The limit of a sum is the sum of the limits.

3. Limit Laws: 2. The limit of a difference is the difference of the limits.

4. Limit Laws: 3. The limit of a constant times a function is the constant times the limit of the function.

5. Evaluate the limit and justify each step. Example 1: Note: If we let f(x)=2x 2 -3x+4, then f(5)=39.

6. Limit Laws: 4. The limit of a product is the product of the limits.

7. Limit Laws: 5. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).

8. Evaluate the limit and justify your answer. Example 2: Note: As in example 1, if we let f(x)=(x 3 +2x 2 -1)/(5-3x), then f(-2)=-1/11

9. Direct Substitution Property: If f is a polynomial or a rational function and a is in the domain of f, then

10. Let’s try example 1 on page 111.

11. Example 4: Find

12. Limit Laws cont: 6. where n is a positive integer

13. Special Limits: 7. 8.

14. More Special Limits: 9. where n is a positive integer 10.

15. More Limit Laws: 11. where n is a positive integer [If n is even, we assume that ]

16. Example 5: Find Remember: The limit is not necessarily what the function is equal to; it is what the function approaches when x gets close to a

17. Example 6: Evaluate: Sometimes we may have to use algebra to take limits.

18. Example 7: Find Here we have to rationalize the numerator in order to find the limit.

19. Theorem: Some limits are best calculated by first finding the left- and right- hand limits. This theorem is a reminder of what we discovered in previous sections.

20. Example 8: Show that

21. Example 9: Prove that does not exist.

22. Example 10: Show that does not exist.

23. Theorem: If f(x) < g(x) when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a, then

24. The Squeeze Theorem: If f(x) < g(x) < h(x) when x is near a (except possibly at a) and then

25. Example 11: By using the Squeeze Theorem show that