5.18. Calculate the limit

$$
\lim _{x \rightarrow \infty}\left(\frac{4 x^{2}-x+1}{2 x^{2}+x+1}\right)^{\frac{x^{3}}{2-x}}
$$
Solution. This limit is not an indeterminate form, since when

$$
x \rightarrow \infty \frac{4 x^{2}-x+1}{2 x^{2}+x+1} \rightarrow 2, \text { and } \frac{x^{3}}{2-x} \sim -x^{2} \rightarrow -\infty
$$

Therefore:

$$
\begin{gathered}
\lim _{x \rightarrow \infty}\left(\frac{4 x^{2}-x+1}{2 x^{2}+x+1}\right)^{\frac{x^{3}}{2-x}}=\lim _{x \rightarrow \infty}\left(2-\frac{3 x-1}{2 x^{2}+x+1}\right)^{\frac{x^{3}}{2-x}}= \\
=\left(\lim _{x \rightarrow \infty}\left(2-\frac{3 x-1}{2 x^{2}+x+1}\right)\right)^{\lim _{x \rightarrow x} \frac{x^{3}}{2-x}}=2^{-\infty}=0
\end{gathered}
$$