Answer
See the explanation
Work Step by Step
a. The vertical asymptotes of $r$ are the lines $x=a$, where $a$ is a zero of the denominator. By factoring and finding zeros of a denominator, we can find the vertical asymptotes of a rational function.
b. Let $s$ be a rational function
$s(x)=\frac{a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}}{b_{m}x^{m}+b_{m-1}x^{m-1}+...+b_{1}x+b_{0}}$.
The Horizontal Asymptotes of $s$ are:
- If the degree of the numerator is greater than the degree of the denominator ($n>m$), then the function has no Horizontal Asymptotes
- If the degree of the numerator equals the degree of the denominator ($n=m$), then the Horizontal Asymptotes is $a_n/b_m$.
- If the degree of the numerator is less than the degree of the denominator, then the Horizontal Asymptotes is the $x$-axis or the line where $y=0$.
c. Given the rational function
$f(x)=\frac{5x^2+3}{x^2-4}$,
the Vertical Asymptotes are:
$x^2-4=0, x^2=4, x= \pm 2$
and the Horizontal Asymptotes are,
$y=\frac{5}{1}=5$,
thus, the Vertical Asymptotes are the lines $x=\pm2$ and the Horizontal Asymptote is the line $y=5$.
