Example 4. Find $\lim _{x \rightarrow 1} \frac{3 x^{2}-x-2}{4 x^{2}-5 x+1}$.
Solution. When $x=1$, both the numerator and the denominator become zero, resulting in an indeterminate form of $\frac{0}{0}$. We will transform the given function by factoring the numerator and the denominator using the formula

$$
a x^{2}+b x+c=a\left(x-x_{1}\right)\left(x-x_{2}\right)
$$

where $x_{1}$ and $x_{2}$ are the roots of the equation $a x^{2}+b x+c=0$.

Substituting the corresponding expressions and canceling the common factor $(x-1) \neq 0$, we get

$$
\begin{gathered}
\lim _{x \rightarrow 1} \frac{3 x^{2}-x-2}{4 x^{2}-5 x+1}=\lim _{x \rightarrow 1} \frac{3(x-1)\left(x+\frac{2}{3}\right)}{4(x-1)\left(x-\frac{1}{4}\right)}= \\
=\lim _{x \rightarrow 1} \frac{3\left(x+\frac{2}{3}\right)}{4\left(x-\frac{1}{4}\right)}=\frac{3\left(1+\frac{2}{3}\right)}{4\left(1-\frac{1}{4}\right)}=\frac{5}{3}
\end{gathered}
$$