I have to prove that a function is either Lebesgue integrable or to give counterexample. In particular, the problem is related to almost direct application of Dominated Convergence theorem.
Prove or disprove $$\lim _{n\rightarrow \infty} \int_{[0,1]}\exp\left(\frac{x^2}{n}\right)dx = \int_{[0,1]} \lim _{n\rightarrow \infty} \exp\left(\frac{x^2}{n}\right)dx.$$
I managed to formulate the following proof:
Looking at the function $f(x) = \exp(\frac{x^2}{n}), x \in [0,1], n \geq 1$. It is obvious that it is dominated by the function $g(x) = \exp(x^2)$, since $\exp(\frac{x^2}{n}) \leq \exp(x^2)$. Moreover, $\int _{[0,1]} e^{x^2} dx< +\infty$. From the above-stated theorem it follows that $f(x)$ is Lebesgue Integrable and $\lim _{n\rightarrow \infty} \int_{[0,1]}\exp(\frac{x^2}{n})dx = \int_{[0,1]} \lim _{n\rightarrow \infty} \exp(\frac{x^2}{n})dx$.
My question here are two. First, I am not sure about the dominance of $e^{x^2}$ on that interval. Second, since the function is dominated, it follows that it is Lebesgue integrable ($g(x)$ is integrable). Does it follow that we can interchange the limits as done in the statement?