Example 1. Find $\lim _{x \rightarrow+\infty}\left(\sqrt{x^{2}+6 x+5}-x\right)$.
Solution. When $x \rightarrow+\infty$, the given function represents the difference of two infinitely large quantities taking positive values (case $\infty-\infty$). By multiplying and dividing the given function by $\left(\sqrt{x^{2}+6 x+5}+x\right)$, we get

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

$$
=\lim _{x \rightarrow+\infty} \frac{6+\frac{5}{x}}{\sqrt{1+\frac{6}{x}+\frac{5}{x^{2}}}+1}=\frac{6}{1+1}=3
$$