# real analysis – How to find the limit of \$ frac{n^2}{7^{sqrt{n}}}\$ in \$+infty\$? I know that it exists and can’t find it with Stolz theorem.

How do I find the limit of: $$frac{n^2}{7^{sqrt{n}}}$$ in $$+infty$$?

I thought about Stolz theorem so I got: $$frac{(n+1)^2-n^2}{7^{sqrt{n+1}}-7^{sqrt{n}}}$$ but it doesn’t seem to lead anywhere (at least not through e and ln).

And I know for fact that there is some limit because:
$$lim_{n to +infty} frac{a_n+1}{a_n} = frac{frac{(n+1)^2}{7^{sqrt{n+1}}}}{frac{n^2}{7^{sqrt{n}}}} = frac{(n+1)^2*7^{sqrt{n}}}{7^{sqrt{n+1}}*n^2} = frac{(n+1)^2*7^{sqrt{n}}}{7^{sqrt{n}sqrt{frac{1}{n}+1}}*n^2} implies frac{(n+1)^2}{n^2}implies 1$$
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