Answer
(a) $\lim_{x\to\infty}f(x)=-3$
(b) $\lim_{x\to-\infty}f(x)=-3$
Work Step by Step
$$f(x)=\frac{2}{x}-3$$
(a) As $x\to\infty$, or $x$ gets infinitely large, $2/x$ will approach $0$.
Therefore, $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}\Big(\frac{2}{x}-3\Big)=0-3=-3$$
(b) As $x\to-\infty$, or $x$ gets infinitely small, $2/x$ will approach $-2/\infty$, and so will approach $0$ as well.
Therefore, $$\lim_{x\to-\infty}f(x)=\lim_{x\to-\infty}\Big(\frac{2}{x}-3\Big)=0-3=-3$$
A graph of the function $f(x)$ is enclosed below, which shows that $f(x)$ approaches $-3$ as $x$ approaches either $\infty$ or $-\infty$.
