Bringing the limit inside the integral with asymptotic equilvance?

I was working on a limit problem and I think I found the solution to it, but I am not sure that what I did is right.
I am working with the following sequence of functions$$ f_{n}(x) = n\, e^{\frac1{n\sqrt x}}-n,$$and I want to compute the limit
$$\lim_{n\to+\infty}\int_{0}^{1}f_{n}(x)\,\mbox{d}x.$$Since $\,e^t-1 \sim t\,$ as $\, t\to 0$, I can say that for all $x \in (0,1)$
$$n(e^{\frac{1}{\sqrt{x}n}}-1)\sim \frac{n}{n\sqrt{x}}=\frac{1}{\sqrt{x}}\,\,\,\,\,\,\, (\mbox{as }n\to+\infty).$$ Can this pointwise convergence let me say that$$\lim_{n\to+\infty}\int_{0}^{1}f_{n}(x)\,\mbox{d}x=\int_{0}^{1}\frac{1}{\sqrt{x}} \,\mbox{d}x \qquad?$$