Answer
$$\lim _{x \rightarrow 2} \frac{2 x^{2}-5 x+2}{5 x^{2}-7 x-6} =\frac{3}{13}$$
Work Step by Step
Given $$\lim _{x \rightarrow 2} \frac{2 x^{2}-5 x+2}{5 x^{2}-7 x-6}$$
using the method of replacement
\begin{align*}
\lim _{x \rightarrow 2} \frac{2 x^{2}-5 x+2}{5 x^{2}-7 x-6}&=\lim _{x \rightarrow 2} \frac{2 (2)^{2}-5(2)+2}{5 (2)^{2}-7 (2)-6}\\
&=\frac{0}{0}
\end{align*}
This is an indeterminate form, then applying L'Hôpital's Rule, we
get
\begin{align*}
\lim _{x \rightarrow 2} \frac{2 x^{2}-5 x+2}{5 x^{2}-7 x-6}&= \lim _{x \rightarrow 2} \frac{4 x-5}{10 x-7 }\\
&= \lim _{x \rightarrow 2} \frac{4(2)-5}{10(2)-7 }\\
&=\lim _{x \rightarrow 2}\frac{3}{13}\\
&=\frac{3}{13}
\end{align*}
