calculus - Find: $lim_{n to infty} int_0^{infty} arctan(nx) e^{- x^n}dx$

Find:

$$\lim_{n \to \infty} \int_0^{\infty} \arctan(nx) e^{- x^n}dx$$

Probably, no recursive form could be found, and elementary tools (integration by parts, change of variable, etc.) are not useful here. How can I find such a limit?

Thank you.

Answer

Start by thinking about pointwise limits. For $x>0$, $\arctan(nx) \to \pi/2$. For $01$, $x^n \to +\infty$. Hence for $01$, $e^{-x^n} \to 0$. So the pointwise limit is $\frac{\pi}{2} \chi_{(0,1)}$, except maybe at $0$ and $1$ which don't matter.

So we might intuitively expect the limit to be $\int_0^1 \frac{\pi}{2} dx = \frac{\pi}{2}$. Try to use an integral convergence theorem, such as the dominated convergence theorem, to justify this result.