How to find the limit of the sequence $x_n=n^{n^{0.5}} - 2^n?$

Question

How to find the limit of the sequence $x_n=n^{n^{0.5}} - 2^n?$

I have tried converting the sequence to exponent of logs but that brings back the indeterminate form.

Answer 1

Note that $2^n=(2^{\sqrt n})^{\sqrt n}\gg n^{\sqrt n}$ for large $n$.

Answer 2

Converting the sequence to exponent of logs is a good strategy: we have that $$n^{n^{0.5}}=\exp(\sqrt{n}\ln(n))\quad\text{and}\quad 2^n=\exp(n\ln(2)).$$ By comparing the exponents, which one grows faster to infinity? Note that $$\lim_{n\to +\infty}\left(n\ln(2)-\sqrt{n}\ln(n)\right)=\lim_{n\to +\infty}n\left(\ln(2)-\frac{\ln(n)}{\sqrt{n}}\right)=+\infty\cdot (\ln(2)-0)=+\infty.$$