the limit of a sequence to another sequence indeterminate form

Question

I am trying to solve this limit question...I have tried to take the natural log of an^(bn), but somehow ended up with infinity times infinity. Is there any way to this? Thanks a lot.

Answer 1

Once $n$ is large enough, $b_n > 1$ and $\lvert a_n\rvert < 1$ so that

$$\lvert a_n^{b_n} \rvert = \lvert a_n\rvert^{b_n} < \lvert a_n\rvert$$

which converges to zero.

Answer 2

You have $$ \lim_{n\to\infty}\log(a_n^{b_n})= \lim_{n\to\infty}b_n\log a_n=-\infty $$