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

2. OBJECTIVES:
SWBAT:
UNDERSTAND AND BE ABLE TO APPLY THE BASIC LIMITS LAWS FOR LIMITS AT INFINITY
DESCRIBE THE LIMITS AT INFINITY
CALCULATE LIMITS AT INFINITY OF A RATIONAL FUNCTION

3. Part I: Infinite Limits:
vertical asymptote at x=0.

4. “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.”

5. 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.

6. 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.

7. 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.

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

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

10. Horizontal Asymptote
Infinite limits describe the asymptotic behavior of a function, which is behavior of the graph as we move out to the right or to the left.
Example 1, pg. 100.

11. For all n>0,
If n is a whole number,

12. EXAMPLES 2-4, pgs

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

14. Example
Divide by

15. More Examples