1. Euclid was a Greek mathematician best known for his treatise on geometry: The Elements. This influenced the development of Western mathematics for more than 2000 years. Euclid was a Greek mathematician best known for his treatise on geometry: The Elements. This influenced the development of Western mathematics for more than 2000 years. Euclid of Alexandria 325 – 265 B.C. Euclid of Alexandria 325 – 265 B.C.

2. So we say, This limit shows that f (x) has a horizontal asymptote at y = 1.

3. We take limits to infinity to analyze end-behavior of functions.

4. Left-Hand Behavior Right-Hand Behavior

6. As the denominator gets larger, the value of the fraction gets closer to zero. So by definition, y = 0 is a horizontal asymptote. Recall vertical asymptote at x = 0. horizontal asymptote at y = 0.

7. Method IMethod II Algebraic Limits

8. Dominating term effect (DTE) we get dominant terms in numerator and denominator

10. Recall: Example

11. This number becomes insignificant as. I call this the dominating term effect. There is a horizontal asymptote at y = 1.

12. Find: so for : Thus, by the Squeeze Theorem: x y When we graph this function, the limit appears to be zero.

13. Find:

16. Recall Infinite Limits: As the denominator approaches zero, the value of the fraction gets very large. If the denominator is positive then the fraction is positive. If the denominator is negative then the fraction is negative. vertical asymptote at x =0.

17. The denominator is positive in both cases, so the limit is the same.

18. a. Find the domain of f (x). b. Determine any asymptotes for f (x).  c. Where is f (x) not continuous?