Evaluate limit of the form $0^{\infty}$

All limits of the form $0^\infty$ have $0$ as limit. This is not an indeterminate form, like $\infty^0$ or $\frac\infty\infty$.

To have some notation and formality, let $a_n\to 0$ and $b_n\to \infty$ be two sequences, and for simplicity assume that $a_n, b_n> 0$. Then $a_n^{b_n}\to 0$. You can show this, for instance, by comparison:

• Option 1: Since $a_n\to 0$ and $b_n\to \infty$, there is some point $N$ where $a_n\leq \frac12$ and $b_n\geq 1$ for all $n\geq N$. Thus we get $$a_n^{b_n}\leq \left(\frac12\right)^{b_n}$$for all $n\geq N$. And the right-hand side certainly goes to $0$.
• Option 2: There is some point $N$ where $a_n\leq 1$ and $b_n\geq 1$ for all $n\geq N$. Then $$ a_n^{b_n} \leq a_n $$ for all $n\geq N$. And the right-hand side certainly goes to $0$.

(These arguments work more or less unchanged for negative or zero $a_n, b_n$ as well, as long as $a_n^{b_n}$ makes sense. But I skipped it as I feel it's more difficult to type corrrecly, more difficult to read, and doesn't actually give that much more value to this post.)