1. AKA, Asymptotes of Rational Functions, Calculus Style
Limits and Infinity
AKA, Asymptotes of Rational Functions,
Calculus Style

2. Part I: Infinite Limits:
vertical asymptote at x=0.
online.math.uh.edu/HoustonACT/Greg_Kelly.../Calc02_2.ppt

3. “On the contrary, it tells HOW the limit FAILS to exist.”
IMPORTANT NOTE:
The equal sign in the statement does NOT mean the limit exists!
“On the contrary, it tells HOW
the limit FAILS to exist.”
Page 80 Dr. Larson & Dr. Hostetler

4. Definition of a Vertical Asymptote
If f(x) approaches infinity or negative infinity as x approaches c from the left or right, then x = c is a vertical asymptote of f.

5. Digging deeper…
Infinity is a very special idea. We know we can't reach it, but we can still try to work out the value of functions that have infinity in them.

6. Question: What is the value of 1/∞ ? Answer: We don't know!
Maybe we could say that 1/∞ = 0, ... but if we divide 1 into infinite pieces and they end up 0 each, what happened to the 1?
In fact 1/∞ is known to be undefined.

7. But We Can Approach It! x 1/x 1 1.00000 2 0.50000 4 0.25000 10 0.10000
1,000
10,000

8. The limit of 1/x as x approaches Infinity is 0 Furthermore:

9. Limits at Infinity Divide through by the highest power of x Simplify
Substitute 0 for 1/xn

10. Example
Divide by

11. More Examples

13. Worksheet Review: Page 91,
Homework
Worksheet
Review: Page 91,
#3-5, 11-18, 27-30, 41, 42, 44, 47, 49, 50, 55, 59-61, 63-64