### Home > CALC > Chapter 2 > Lesson 2.2.4 > Problem2-100

2-100.

Let $f(x) = x^2-9$, and $g(x) = 2x^2 − 12x + 18$. Find all horizontal asymptotes, vertical asymptotes, and holes (if any) for $y = \frac { f ( x ) } { g ( x ) }$ and$y = \frac { g ( x ) } { f ( x ) }$.

$y=\frac{f(x)}{g(x)}=\frac{x^{2}-9}{2x^{2}-12x + 18}$

Find horizontal asymptotes by using limits as $x → ±∞$.

$\lim\limits_{x\rightarrow \infty }\frac{x^{2}-9}{2x^{2}-12x+18}=\frac{1}{2}$
$\lim\limits_{x\rightarrow -\infty }\frac{x^{2}-9}{2x^{2}-12x+18}=\frac{1}{2}$
Horizontal asymptote at $y=\frac{1}{2}$.

Find vertical asymptotes and holes by factoring and simplifying.
$\frac{x^{2}-9}{2x^{2}-12x+18}=\frac{(x+3)(x-3)}{2(x+3)(x-3)}$
Even though $(x-3)$ cancels out, there remains an $(x-3)$ in the denominator. No holes, there is a vertical asymptote at $x = 3$.

$y=\frac{g(x)}{f(x)}=\frac{2x^{2}-12x+18}{x^{2}-9}$

Find horizontal asymptotes by using limits as $x → ±∞$.

Find vertical asymptotes and holes by factoring and simplifying.