What is $\lim_{n\rightarrow \infty} e^{-x^n}$?
More precisely I found this example in the book, and it is written that final answer is 1. However it does not give any hints about how to solve. $$\lim_{n\rightarrow \infty}\int_0^n e^{-x^n} dx$$ I know that it is possible to change limit and integral. I mean the order, if I find $g(x)$ such that $|f_n(x)|\lt g(x)$. However I do not know in this case how to find such $g$ as well.
You can take $$g(x) := \begin{cases} 1 & \text{if $x \in [0,1]$,} \\ e^{-x} & \text{if $x > 1$.} \end{cases}$$ Note $$x>1 \implies x^n \to \infty \implies e^{-x^n} \to 0,$$ and $$x \in [0,1] \implies x^n \to 0 \implies e^{-x^n} \to 1.$$ Now, exchange limits by the Dominated Convergence Theorem