Answer
$$\frac{1}{2}$$
Work Step by Step
Given $$\lim _{x \rightarrow \infty} \frac{3 x^{4}+x-5}{6 x^{4}-2 x^{2}+1} $$
Then
\begin{align*}
\lim _{x \rightarrow \infty} \frac{3 x^{4}+x-5}{6 x^{4}-2 x^{2}+1}&=\lim _{x \rightarrow \infty} \frac{3+\frac{1}{x^{3}}-\frac{5}{x^{4}}}{6-\frac{2}{x^{2}}+\frac{1}{x^{4}}}\\
&= \frac{3+0+0}{6-0+0}\\
&=\frac{1}{2}
\end{align*}