Answer
(a) $0$
(b) $0$
Work Step by Step
(a) \begin{align*}
\lim_{x\to \infty}g(x)&= \lim_{x\to \infty} \frac{10x^5+x^4+31}{x^6}\\
&= \lim_{x\to \infty} \frac{10x^5/x^6+x^4/x^6+31/x^6}{x^6/x^6}\\
&= \frac{ \lim_{x\to \infty}(10/x)+ \lim_{x\to \infty}(1/x^2)+ \lim_{x\to \infty}(31/x^6)}{ \lim_{x\to \infty}(1)}\\
&= \frac{0+0+0}{1}\\
&=0
\end{align*}
(b)\begin{align*}
\lim_{x\to -\infty}g(x)&= \lim_{x\to- \infty} \frac{10x^5+x^4+31}{x^6}\\
&= \lim_{x\to- \infty} \frac{10x^5/x^6+x^4/x^6+31/x^6}{x^6/x^6}\\
&= \frac{ \lim_{x\to- \infty}(10/x)+ \lim_{x\to -\infty}(1/x^2)+ \lim_{x\to- \infty}(31/x^6)}{ \lim_{x\to- \infty}(1)}\\
&= \frac{0+0+0}{1}\\
&=0
\end{align*}
