find all horizontal asymptotes of the following function.\n\n$f(x)=\\frac{2(3x + 5)(3x - 5)}{x…

find all horizontal asymptotes of the following function.\n\n$f(x)=\\frac{2(3x + 5)(3x - 5)}{x - 9}$\n\nanswer attempt 1 out of 2\n\none horizontal asymptote\nno horizontal asymptotes\none horizontal asymptote\ntwo horizontal asymptotes

find all horizontal asymptotes of the following function.\n\n$f(x)=\\frac{2(3x + 5)(3x - 5)}{x - 9}$\n\nanswer attempt 1 out of 2\n\none horizontal asymptote\nno horizontal asymptotes\none horizontal asymptote\ntwo horizontal asymptotes

Answer

Explanation:

Step1: Expand the numerator

First, expand (2(3x + 5)(3x - 5)) using the difference - of - squares formula ((a + b)(a - b)=a^{2}-b^{2}). Here (a = 3x) and (b = 5), so ((3x + 5)(3x - 5)=9x^{2}-25). Then (2(3x + 5)(3x - 5)=18x^{2}-50). So the function becomes (f(x)=\frac{18x^{2}-50}{x - 9}).

Step2: Use the rule for horizontal asymptotes

For a rational function (y=\frac{f(x)}{g(x)}=\frac{a_{n}x^{n}+\cdots+a_{0}}{b_{m}x^{m}+\cdots+b_{0}}), if (n>m), there is no horizontal asymptote. In our function (n = 2) (degree of the numerator (18x^{2}-50)) and (m = 1) (degree of the denominator (x - 9)). Since (n=2>m = 1)

Answer:

No Horizontal Asymptotes