is it true that $\lim_{n \to \infty} {n^{2\log_{2}(n)-4}} = \infty$?

Question

I was computing the following limit: $\lim_{n \to \infty} {n^{2\log_{2}(n)-4}} = \infty$

and I am obtaining $\infty^\infty$. Is it equal to $\infty?$

Answer 1

Yes, it is equal to infinity. The equation is $n^{2log_2(n)-4}$ As you mentioned, this is equal to $\infty^{\infty}$, which is infinity. This is because infinity to the power of infinity is infinity. For example, take any number over 1). Raising it to the power of itself will always produce a bigger number, e.g. $10^10$ is ten billion, and this holds true for any number greater than 1. Since infinity is obviously greater than one, infinity to the power of infinity is greater than infinity. However, since nothing is greater than infinity, infinity to the power of infinity is infinity. Therefore, The answer is infinity

Answer 2

Yes that's right, since as noticed $\infty^\infty$ is not an indeterminate form, indeed in this case we have that as $2\log_{2}(n)-4\ge1$

$$n^{2\log_{2}(n)-4}\ge n\to \infty$$