This document defines and provides examples of vertical and horizontal asymptotes of functions. It states that a vertical asymptote occurs when the function approaches infinity as x approaches a particular value, making that x-value an asymptote. A horizontal asymptote occurs when the limit of the function is a real number L as x approaches positive or negative infinity, making the lines y=L asymptotes. It provides properties and examples of determining horizontal asymptotes of rational functions based on the degrees of the numerator and denominator.

1. 3.6 Asymptotes:Goal: to find the asymptotes of functions to find infinite limits of functionsDef: Vertical AsymptoteIf 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 ƒEx. 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 AsymptoteIf ƒ(x) is a function and L1 and L2 are real numbers, the statementsand- Denotes limits at infinity, the lines y = L1 and y = L2 are horizontal asymptotes of ƒ(x)

5. PropertiesFor all r > 0For all r > 0

6. Example

7. From our prior studies of rational functionswe learned the following about horizontal asymptotesHorizontal Asymptotes of Rational Functions:Let: be a rational function1. 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.