Why $\lim_{n\to \infty} \frac{(2^2)^{\sqrt n}}{(1+(10)^{-2^2})^n}$ has no limit?
$\lim_{n\to \infty} \frac{(2^2)^{\sqrt n}}{(1+(10)^{-2^2})^n}$
I looked it up in the wolfram and my intuition tells me that this expression has no limit, but every step I did doesn't lead me to the solution.
Observe that for $n$ big enough we have
$$5<(1+10^{-4})^\sqrt{n}$$ so $$5^{\sqrt{n}}<(1+10^{-4})^n $$ and then $$\frac{4^{\sqrt n}}{(1+10^{-4})^n} <\frac{4^{\sqrt n}}{5^{\sqrt{n}}}=\left(\frac{4}{5}\right)^{\sqrt n} \to 0$$