Compute
$\lim_{n\to\infty} \int^{1}_{0}\left (x+\frac{x^2}{n+e^x}\right )\,dx$
With reasoning
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I have come to the conclusion that what I am required to do is to show that the integrand converges uniformly on $(0,1)$, meaning that it will in turn be Riemann integrable and we can use the following fact which holds if this criteria is met:
$\lim_{n\to\infty} \int^{1}_{0}x+\frac{x^2}{n+e^x}dx=\int^{1}_{0}\lim_{n\to\infty} (x+\frac{x^2}{n+e^x})dx$
How then would I go about proving that the function $(x+\frac{x^2}{n+e^x})$ is uniformly convergent on $(0,1)$?