Evaluating certain limits of integrals

Question

I'm dealing with the following limits:

$$ 1.) \lim_{n \rightarrow \infty} \int_{-1}^{1} e^{\frac{x^2}{n}} dx $$

$$ 2.) \lim_{n \rightarrow \infty} \int_{1}^{3} \frac{nx^2 +3}{x^3+nx} dx $$

$$ 3.) \lim_{n \rightarrow \infty} \int_{0}^{\frac{\pi}{2}} (\sin{\frac{x}{n}}+\cos{\frac{x}{n}})^{\frac{1}{2}} dx $$

I assumed that all the sequence of functions all converge uniformly so that the limit and the integral can be interchanged. So, the first integral would be $2$ as $e^{\frac{x^2}{n}} \rightarrow 0 $ as $n \rightarrow \infty $. Similarly, the other two integrals would be $4$ and $\frac{\pi}{2}$ respectively. Am I correct in making these assumptions? Are my results correct?

Answer 1

You can interchange limit and integral with the help of the DCT. Choose for

1. $g(x)=e^{x^2}$
2. $g(x)=(nx^2+3)/n$
3. $g(x)=\sqrt{2}$

These functions are all integrable in their given domain. So yes it is possible to interchange limit and integral in this three cases.