Answer
(a) $0$
(b) $0$
Work Step by Step
(a) \begin{align*}
\lim_{x\to \infty} f(x)&= \lim_{x\to \infty} \frac{ 3x+7}{x^2-2}\\
&= \lim_{x\to \infty} \frac{ 3x/x^2+7/x^2}{x^2/x^2-2/x^2}\\
&=\frac{ \lim_{x\to \infty} (3 /x) + \lim_{x\to \infty} (7/x^2)}{ \lim_{x\to \infty} (1)- \lim_{x\to \infty} (2/x^2)}\\
&=\frac{0+0}{1-0}\\
&=0
\end{align*}
(b) \begin{align*}
\lim_{x\to -\infty} f(x)&= \lim_{x\to- \infty} \frac{ 3x+7}{x^2-2}\\
&= \lim_{x\to- \infty} \frac{ 3x/x^2+7/x^2}{x^2/x^2-2/x^2}\\
&=\frac{ \lim_{x\to -\infty} (3 /x) + \lim_{x\to- \infty} (7/x^2)}{ \lim_{x\to- \infty} (1)- \lim_{x\to -\infty} (2/x^2)}\\
&=\frac{0+0}{1-0}\\
&=0
\end{align*}