Answer
$$\infty$$
Work Step by Step
Given
$$\lim _{x \rightarrow \infty} \frac{x^{4}-3x^2+x}{x^{3}-x+2}$$
Then
\begin{aligned}
\lim _{x \rightarrow \infty} \frac{x^{4}-3x^2+x}{x^{3}-x+2} &=
\lim _{x \rightarrow \infty} \frac{x^{4}/x^{3}-3x^2/x^{3}+x/x^{3}}{x^{3}/x^{3}-x/x^{3}+2/x^{3}}\\
&= \lim _{x \rightarrow \infty} \frac{x -3 /x +1/x^{2}}{1-1/x^{2}+2/x^{3}}\\
&= \frac{ \lim _{x \rightarrow \infty}x - \lim _{x \rightarrow \infty}3 /x + \lim _{x \rightarrow \infty}1/x^{2}}{ \lim _{x \rightarrow \infty}1- \lim _{x \rightarrow \infty}1/x^{2}+ \lim _{x \rightarrow \infty}2/x^{3}}\\
&= \lim _{x \rightarrow \infty}x\\
&=\infty
\end{aligned}
