Answer
$$4$$
Work Step by Step
Given $$ \lim _{x\to \infty } \frac{\left(2x^2+1\right)^2}{\left(x-1\right)^2\left(x^2+x\right)} $$
\begin{aligned}
\lim _{x\to \infty } \frac{\left(2x^2+1\right)^2}{\left(x-1\right)^2\left(x^2+x\right)} &= \lim _{x\to \infty }\frac{4x^4+4x^2+1}{x^4-x^3-x^2+x}\\
&= \lim _{x\to \infty }\frac{\frac{4x^4}{x^4}+\frac{4x^2}{x^4}+\frac{1}{x^4}}{\frac{x^4}{x^4}-\frac{x^3}{x^4}-\frac{x^2}{x^4}+\frac{x}{x^4}}\\
&=\lim _{x\to \infty }\frac{4+\frac{4}{x^2}+\frac{1}{x^4}}{1-\frac{1}{x }-\frac{1}{x^2}+\frac{1}{x^3}}\\
&=\lim _{x\rightarrow \infty}\frac{4+0+0}{1-0-0+0}\\
&=4
\end{aligned}
