Compute $\lim_{n \rightarrow \infty} \int_0^1 \frac{n+n^kx^k}{(1+\sqrt{x})^n}$, where $k \geq 1$.
Letting $f_n(x) := \frac{n+n^kx^k}{(1+\sqrt{x})^n} $, we have the pointwise limit $f_n \rightarrow 0$ a.e. . To see this, consider the first term, $\lim_{n \rightarrow \infty} \frac{n}{(1+\sqrt{x})^n} = \lim_{n \rightarrow \infty} \frac{1}{\log(1+\sqrt{x})(1+\sqrt{x})^{n}} = 0$ by L'Hopital, for $x \in (0,1)$. We can show similarly for the second term $\frac{n^kx^k}{(1+\sqrt{x})^n}$.
However, I am unable to find an integrable dominating function for the first term, $\frac{n}{(1+\sqrt{x})^n}$. By simulation, however, it seems like $\int_0^1 \frac{n}{(1+\sqrt{x})^n} \rightarrow 0$, and moreover that it is dominated by $\frac{1}{\sqrt{x}}$, but I'm unable to show it.
I'd also appreciate learning any general strategies for finding bounds that can be specialized for this question. I'm guessing a lot of it is trial and error, but my trouble is that all I'm able to use is that $(1+u)^a \geq 1+u^a$ for the above case, which only yields $\frac{n}{(1+\sqrt{x})^n} \leq \frac{n}{1+x^{n/2}}$, and from where I am unable to see how to proceed.