Question

graph all vertical and horizontal asymptotes of the rational
$f(x)=\frac{x^{2}-4}{-2 x^{2}-5}$

Explanation:

Step1: Find the vertical asymptote

Set the denominator equal to zero: $-2x^{2}-5 = 0$.
Rearrange it: $2x^{2}=-5$. Since $x^{2}\geq0$ for all real $x$, there is no real solution. So, there is no vertical asymptote.

Step2: Find the horizontal asymptote

For a rational function $f(x)=\frac{a_nx^n+\cdots+a_0}{b_mx^m+\cdots+b_0}$, when $n = m$ (here $n = m = 2$), the horizontal asymptote is $y=\frac{a_n}{b_m}$.
Here $a_n = 1$ (coefficient of $x^{2}$ in the numerator) and $b_m=-2$ (coefficient of $x^{2}$ in the denominator). So, the horizontal asymptote is $y =-\frac{1}{2}$.

Answer:

The horizontal asymptote is $y =-\frac{1}{2}$ and there is no vertical asymptote.