Notes for section 3.6

1. 3.6 Asymptotes:Goal: to find the asymptotes of functions to find infinite limits of functions Def: Vertical Asymptote If ƒ(x) approaches infinity (or negative infinity) as x->c from the right or the left, then the line x = c is a vertical asymptote of ƒ Ex. Graph Notice that as x->2- Notice that as x->2+

2. Now Ex. Graph Notice that as x-> -3- Notice that as x-> -3+

3. Possible Vertical Asymptotes correspond to x-values that make the denominator zero.Try the following examples:

4. Def: Horizontal Asymptote If ƒ(x) is a function and L1 and L2 are real numbers, the statements and - Denotes limits at infinity, the lines y = L1 and y = L2 are horizontal asymptotes of ƒ(x)

5. Properties For all r > 0 For all r > 0

6. Example

7. From our prior studies of rational functionswe learned the following about horizontal asymptotes Horizontal Asymptotes of Rational Functions:Let: be a rational function 1. If the degree of the numerator is less than the degree of the denominator then y = 0 is a horizontal asymptote. In this case the numerator is 1st degree and the denominator is 2nd degree. Since the degree of the numerator is less than the degree of the denominator then y = 0 is a horizontal asymptote. That also means that…

8. 2. If the degree of the numerator is equal to the degree of the denominator then a horizontal asymptote. a1 and a2 are the leading coefficients of p(x) and q(x) In this case both the numerator and the denominator are 2nd degree. 3 is the leading coefficient of the numerator and 4 is the leading coefficient of the denominator. Be careful here, to find the leading coefficient, both the numerator and denominator must be in standard form (descending order of exponents)! y = ¾ is our horizontal asymptote.

9. 3. If the degree of the numerator is greater than the degree of the denominator then graph of ƒ(x) has no horizontal asymptote. As x-> -∞ or as x-> ∞ the function is unbounded.