1. Infinite number of rooms Hotel Paradox Proofs
1/3=
2inf = inf implies inf = 0
Intro to Calculus
Rainbow Bridge, find the area under the curve
Find the instantaneous rate of change or tangent line at an exact point.

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

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

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.”
Page 80 Dr. Larson & Dr. Hostetler

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

11. Example
Divide by

12. More Examples