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- 1. 3.6 Asymptotes:Goal: to find the asymptotes of functions to find infinite limits of functions<br />Def: Vertical Asymptote<br />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<br />Ex. Graph <br />Notice that as x->2-<br />Notice that as x->2+<br />
- 2. Now <br />Ex. Graph <br />Notice that as x-> -3-<br />Notice that as x-> -3+<br />
- 3. Possible Vertical Asymptotes correspond to x-values that make the denominator zero.Try the following examples:<br />
- 4. Def: Horizontal Asymptote<br />If ƒ(x) is a function and L1 and L2 are real numbers, the statements<br />and<br />- Denotes limits at infinity, the lines y = L1 and y = L2 are horizontal asymptotes of ƒ(x) <br />
- 5. Properties<br />For all r > 0<br />For all r > 0<br />
- 6. Example<br />
- 7. From our prior studies of rational functionswe learned the following about horizontal asymptotes<br />Horizontal Asymptotes of Rational Functions:Let: be a rational function<br />1. If the degree of the numerator is less than the degree of the denominator then y = 0 is a horizontal asymptote.<br />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…<br />
- 8. 2. If the degree of the numerator is equal to the degree of the denominator then a horizontal asymptote.<br />a1 and a2 are the leading coefficients of p(x) and q(x)<br />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)!<br />y = ¾ is our horizontal asymptote.<br />
- 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.<br />As x-> -∞ or as x-> ∞ the function is unbounded.<br />

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