1. Sec 4: Limits at Infinity

2. Definition of a Horizontal Asymptote
The line y = L is a horizontal asymptote of the graph of f if or

3. Ex 1: Find the limits numerically and then state the horizontal asymptote(s)-∞
-100
-10 -1 1 10
100 ∞
f(x) ?
Horizontal Asymptote(s):

4. Theorem: Limits at Infinity
If r is a positive rational number and c is any real number, then
Furthermore, if is defined when x < 0, then

5. Ex 2: Evaluate a simple Limit at Infinity

6. Ex 3: Evaluate

7. Shortcuts for finding Limits at Infinity
If the degree of the numerator is equal to the degree of the denominator, then the limit is the ratio of the leading coefficients.
If the degree of the numerator is less than the degree of the denominator, then the limit is 0.
If the degree of the numerator is more than the degree of the denominator, then the limit is ∞.

8. Ex 3: Evaluate using the shortcuts for limits at infinityB. C.

9. HOMEWORK
Pg 199 #

10. Ex 4: Limits of Non-Rational Functions
With Non-Rational functions, sometimes there will be 2 horizontal asymptotes, which means two different limits. A. B.
*for -∞, change bottom fraction to

11. Ex 5: Limits at Infinity using Reasoning
A. C. B. D.

12. HOMEWORK
Pg 199 #7-12 (do algebraically)