### 5.17. Compute the limit

$$
\lim _{x \rightarrow \infty}\left(\frac{x^{3}+1}{x^{3}-1}\right)^{x^{3}}
$$
Solution. The calculation of this limit is also related to the resolution of an indeterminate form of type $1^{x}$. We will carry out the transformations recommended in section 5.7.5 (point 5). We have

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

Here we used the continuity of the composition of continuous functions.