1. Section 4.4 Limits at Infinity; Horizontal Asymptotes AP Calculus November 2, 2009 Berkley High School, D1B1 todd1@toddfadoir.com

2. Calculus, Section 4.4, Todd Fadoir2 Limit at Infinity How do we find the value of a function at a point that doesn’t exist? We truly can’t, so we use the idea of approaching infinity as compared to being at infinity.

3. Calculus, Section 4.4, Todd Fadoir3 Limit at Infinity The limit of f(x) as x approaches infinity equals L means that the values of f(x) can be made arbitrarily close to L by making x sufficiently large. How close? Calculus close.

4. Calculus, Section 4.4, Todd Fadoir4 Limit at Negative Infinity The limit of f(x) as x approaches negative infinity equals L means that the values of f(x) can be made arbitrarily close to L by making x sufficiently large and negative.

5. Calculus, Section 4.4, Todd Fadoir5 Horizontal Asymptote

6. Calculus, Section 4.4, Todd Fadoir6 Limits that go to zero

7. Calculus, Section 4.4, Todd Fadoir7 Example Attack 1: Common Sense As x get really big, the “-1” and the “+1” are so small they we can forget about them. We are left with x 2 /x 2. This equals 1. The problem with this attack is that, while it make sense, what rules of algebra did we follow? We put this “proof” in a two column form, what reasons would we give?

8. Calculus, Section 4.4, Todd Fadoir8 Example Attack 2: Algebraic Conclusion: there is a horizontal asymptote at y =1

9. Calculus, Section 4.4, Todd Fadoir9 Example

10. Calculus, Section 4.4, Todd Fadoir10 Example

11. Calculus, Section 4.4, Todd Fadoir11 Assignment Section 4.4, 1-35, 39, odd