TITLE: Evaluation of $\lim_{n\to\infty} \int_0^1 \frac{e^{\displaystyle x^{n}}}{1+x^2}\,\mathrm{d}x$
QUESTION [5 upvotes]: Evaluation of 
$$\lim_{n\to\infty} \int_0^1 \frac{e^{\displaystyle x^{n}}}{1+x^2}\,\mathrm{d}x$$
Sis.

REPLY [2 votes]: I'm might be missing something, but the integral is bounded by:
$$\int_0^1\frac{e}{1+x^2}dx > \int_0^1\frac{e^{x^n}}{1+x^2}dx > \int_0^1\frac{1}{1+x^2}dx$$
I think this is enough to interchange the limit with the integral due to the Dominated convergence theorem, arriving at:
$$\int_0^1\frac{1}{1+x^2}dx = \pi/4$$