1. EVALUATING LIMITS ANALYTICALLY (1.3)
September 6th, 2017

2. ***Recall that the first method of evaluating a limit should always be _________________________________!!!

3. I. PROPERTIES OF LIMITS Thm. 1.1: Some Basic Limits
Let b and c be real numbers and n be a positive integer. 1) 2) 3)

4. Thm. 1.2: Properties of Limits:
Let b and c be real numbers, n be a positive integer, and f and g be functions with the following limits.
1) Scalar multiple:
2) Sum or difference:
3) Product:
4) Quotient:
5) Power:

5. Thm. 1.3: Limits of Polynomial & Rational Functions:
If p is a polynomial function and c is a real number, then .
If r is a rational function given by
and c is a real number such that , then

6. Thm. 1.4: The Limit of a Function Involving a Radical:
Let n be a positive integer.
This limit is valid for all values of c if n is odd, but is only valid for c>0 if n is even.

7. Thm. 1.5: The Limit of a Composite Function:
If f and g are functions such that
and , then

8. Thm. 1.6: Limits of Trigonometric Functions:
Let c be a real number in the domain of the given trigonometric function.
1) )
3) )
5) )

9. Summary: Direct substitution can be used to evaluate limits of all polynomial functions, rational functions, radical functions in which the limits are valid, trigonometric functions, and compositions of each of the aforementioned functions.

10. II. A STRATEGY FOR FINDING LIMITS
Thm. 1.7: Functions that Agree on All but One Point (Functions with a Hole):
Let c be a real number and f(x)=g(x) for all in an open interval containing c. If the limit of g(x) as x approaches c exists, then the limit of f(x) also exists and .

11. A STRATEGY FOR FINDING LIMITS
1) Recognize which limits can be evaluated by direct substitution.
2) If the limit as x approaches c cannot be evaluated by direct substitution, try the dividing out or rationalizing technique to apply theorem 1.7.
3) Use a graph or table to reinforce your conclusion.

12. Ex. 1: Find each limit (if it exists).

13. III. THE SQUEEZE THEOREM
Thm. 1.8: The Squeeze Theorem:
If for all x in an open interval containing c, except possibly at c itself, and if ,
then exists and is equal to L.
Thm. 1.9: Two Special Trigonometric Limits:

14. Proof of Theorem 1.9a: The Special Trigonometric Limit
Consider a sector in the unit circle. Compare the area of the sector, the inscribed triangle, and the larger right triangle that contains the sector.

15. Ex. 2: Find each limit. a. b. c.