Evaluating $\displaystyle \lim _{a\rightarrow \infty }\int _{0}^{1} a^{x} x^{a} dx.$
Let $\displaystyle \epsilon >0$ be given.
$\displaystyle x\mapsto a^{x}$ is continuous at 1 so there is a $\displaystyle d\in ( 0,1)$ such that $\displaystyle |a^{x} -a|< \epsilon $ for all $\displaystyle x\in [ d,1]$.
$\displaystyle |\int _{0}^{1} x^{a} a^{x} \ dx-\ \int _{0}^{1} \ ax^{a} |=|\int _{0}^{1}\left( a^{x} -a\right) x^{a} \ dx|\leq |\int _{0}^{d}\left( a^{x} -a\right) x^{a} \ dx|+|\int _{d}^{1}\left( a^{x} -a\right) x^{a} \ dx|$