Example 6. Find $\lim _{x \rightarrow+\infty} \frac{6 x^{2}+5 x+4}{3 x^{2}+7 x-2}$.
Solution. As $x \rightarrow+\infty$, the numerator and denominator increase without bound (we get an indeterminate form of $\frac{\infty}{\infty}$). To find the limit, we transform the given fraction by dividing its numerator and denominator by $x^{2}$, i.e., by the highest power of $x$. Using the properties of limits, we get

$$
\begin{gathered}
\lim _{x \rightarrow+\infty} \frac{6 x^{2}+5 x+4}{3 x^{2}+7 x-2}=\lim _{x \rightarrow+\infty} \frac{6+\frac{5}{x}+\frac{4}{x^{2}}}{3+\frac{7}{x}-\frac{2}{x^{2}}}=\frac{\lim _{x \rightarrow+\infty}\left(6+\frac{5}{x}+\frac{4}{x^{2}}\right)}{\lim _{x \rightarrow+\infty}\left(3+\frac{7}{x}-\frac{2}{x^{2}}\right)}= \\
=\frac{6+0+0}{3+0-0}=2
\end{gathered}
$$

Here it is taken into account that

$$
\lim _{x \rightarrow+\infty} \frac{c}{x}=0, \lim _{x \rightarrow+\infty} \frac{c}{x^{2}}=0(c=\text { const })
$$

(See formula (1.54)).