<s>[INST] Alternative solutions to $\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{ 1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\mathrm{d}x$ Here is a limit that can be computed directly by performing the integration and then taking
the limit, but the way is rather ugly. What else can we do? Might we avoid the integration? 
$$\lim_{n\to\infty} \frac{1}{\sqrt{n}}\int_{ 1/{\sqrt{n}}}^{1}\frac{\ln(1+x)}{x^3}\mathrm{d}x$$
 [/INST] Close to $x = 0$, we have
$$\frac{\log(1+x)}{x^3} \sim \frac{1}{x^2} - \frac{1}{2x} + \cdots$$
Pick a $\delta > 0$ such that the error term in R.H.S is $O(1)$ on $(0,\delta)$, we have:
$$\begin{align} \int_{\frac{1}{\sqrt{n}}}^1 \frac{\log(1+x)}{x^3} dx 
&= \left(\int_{\frac{1}{\sqrt{n}}}^\delta + \int_{\delta}^1\right) \frac{\log(1+x)}{x^3} dx\\
&= \left[ -\frac{1}{x} -\frac12 \log x \right]_{\frac{1}{\sqrt{n}}}^\delta + O(\delta) + \int_{\delta}^1 \frac{\log(1+x)}{x^3} dx\\
&= \sqrt{n} +\frac12 \log ( \sqrt{n} ) + O(\frac{1}{\delta})
\end{align}$$
Notice $\delta$ has be chosen independent of $n$, we have:
$$\lim_{n\to\infty} \frac{1}{\sqrt{n}} \int_{\frac{1}{\sqrt{n}}}^1 \frac{\log(1+x)}{x^3} dx
= \lim_{n\to\infty} \frac{1}{\sqrt{n}} \left(\sqrt{n} +\frac12 \log ( \sqrt{n} ) + O(\frac{1}{\delta})\right) = 1$$
</s>