5.9. Calculate the limit

$$
\lim _{x \rightarrow \pm \infty}\left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right)
$$
Solution. In this case, we have an indeterminate form of type «$\infty - \infty$». To resolve it, we multiply and divide the given expression by its conjugate. We get

$$
\begin{gathered}
\lim _{x \rightarrow \pm \infty}\left(\sqrt{x^{2}+1}-\sqrt{x^{2}-1}\right)=\lim _{x \rightarrow \pm \infty} \frac{x^{2}+1-x^{2}+1}{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}= \\
=\lim _{x \rightarrow \pm \infty} \frac{2}{\sqrt{x^{2}+1}+\sqrt{x^{2}-1}}=0
\end{gathered}
$$