1. Miss Battaglia AB/BC Calculus
1.3 Evaluating Limits Analytically Objective: Evaluate a limit using properties of limits
Miss Battaglia
AB/BC Calculus

2. Properties of Limits
Remember that the limit of f(x) as x approaches c does not depend on the value of f at x=c… But it might happen!
Direct substitution
substitute x for c
These functions are continuous at c

3. Properties of Limits Example: Evaluating Basic Limits a) b) c)
Theorem 1.1 Some Basic Limits
Let b and c be real numbers and let n be a positive integer.
Example: Evaluating Basic Limits
a) b) c)

4. Theorem 1.2: Properties of Limits
Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.
and
Scalar Multiple:
Sum or difference:
Product:
Quotient:
Power:

5. The Limit of a Polynomial

6. Theorem 1.3: Limits of Polynomial and Rational Functions
If p is a polynomial function and c is a real number, then
If r is a rational function given by r(x)=p(x)/q(x) and c is a real number such that q(c)≠0, then

7. The Limit of a Rational Function
Find the limit:

8. Theorem 1.4: The Limit of a Rational Function
Let n be a positive integer. The following limit is valid for all c if n is odd, and is valid for c>0 if n is even.
Theorem 1.5: The Limit of a Composite Function
If f and g are functions such that
and , then

9. Limit of a Composite Function
f(x)=x2 + 4 and g(x)=
Find

10. Theorem 1.6: Limits of Trigonometric Functions
Let c be a real number in the domain of the given trigonometric function.
Examples: a. b. c.

11. Find the limit: Theorem 1.7: Functions that Agree at All But One Point
Let c be a real number and let f(x)=g(x) for all x≠c 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
Find the limit:

12. Dividing Out Technique
Find the limit:

13. Rationalizing Technique
Find the limit:

14. Theorem 1.8: The Squeeze Theorem
If h(x) < f(x) < g(x) for all x in an open interval containing c, except possibly at c itself, and if
then exists and is equal to L.
Theorem 1.9: Two Special Trig Limits

15. Extra Example

16. A Limit Involving a Trig Function
Find the limit:

17. A Limit Involving a Trig Function
Find the limit:

18. Classwork/Homework Read 1.3
Page 67 #17-75 every other odd, , 107, 108,