TITLE: How to show that $\lim_{n\rightarrow \infty }{a_n}^{b_n}=\alpha ^\beta $?
QUESTION [5 upvotes]: If $\lim_{n\rightarrow \infty }{a_n}=\alpha (\neq 0) $ and $\lim_{n\rightarrow \infty }{b_n}=\beta$, then $\lim_{n\rightarrow \infty }{a_n}^{b_n}=\alpha ^\beta $?
I unconsciously used this but I realized I'd never seen this theorem before. Is it true?

REPLY [6 votes]: Notice that
$$a_n^{b_n}=e^{b_n\log(a_n)}$$
so by the continuity of the exponential and logarithmic functions you have the result, of course with the assumption $\boldsymbol{a_n>0}$ and $\boldsymbol{\alpha>0}$.