Answer
$$\lim_{x\to\infty}h(x)=\lim_{x\to-\infty}h(x)=-\frac{5}{3}$$
Work Step by Step
$$h(x)=\frac{-5+(7/x)}{3-(1/x^2)}$$
(a) As $x\to\infty$, $x^2$ also approaches $\infty$ and therefore, both $1/x^2$ and $7/x$ will approach $0$.
Therefore, $$\lim_{x\to\infty}h(x)=\lim_{x\to\infty}\frac{-5+(7/x)}{3-(1/x^2)}=\frac{-5+0}{3-0}=-\frac{5}{3}$$
(b) As $x\to-\infty$, $x^2$ will approach $\infty$ and therefore, both $1/x^2$ and $7/x$ still approaches $0$.
Therefore, $$\lim_{x\to-\infty}h(x)=\lim_{x\to-\infty}\frac{-5+(7/x)}{3-(1/x^2)}=\frac{-5+0}{3-0}=-\frac{5}{3}$$
A graph of the function $h(x)$ is enclosed below, which shows that $h(x)$ approaches $-5/3$ as $x$ approaches either $\infty$ or $-\infty$.
