<s>[INST] Limit at infinity involving $e$ I am to find the limit of $$\lim_{x \to \infty} \left(1+\frac{x}{5x^3+x^2+8}\right)^ {\dfrac{x^3+8}{x}}$$ 
I could not find the proper substitution here. I would be happy if someone could shed some light. Thanks.
 [/INST] $$
\begin{align}
\lim_{x\to\infty}\left(1+\frac{x}{5x^3+x^2+8}\right)^{\frac{x^3+8}{x}}
&=\lim_{x\to\infty}\left(1+\frac{1}{5x^2+x+8/x}\right)^{x^2+8/x}\\
&=\lim_{x\to\infty}\left(1+\frac{1}{5x^2+x+8/x}\right)^{(5x^2+x+8/x)\frac{x^2+8/x}{5x^2+x+8/x}}\\
&=\left(\lim_{x\to\infty}\left(1+\frac{1}{5x^2+x+8/x}\right)^{(5x^2+x+8/x)}\right)^{\lim\limits_{x\to\infty}\frac{x^2+8/x}{5x^2+x+8/x}}\\[6pt]
&=e^{1/5}
\end{align}
$$
</s>