sequences and series - Calculate $lim_{nto infty}(n+1)^{frac{1}{sqrt{n}}}$

Calculate the following limit:

$$\lim_{n\to \infty}(n+1)^{\frac{1}{\sqrt{n}}}$$

I have tried to use the squeeze theorem and other convergence tests but all failed.

Please, any help?

Answer

Note that

$$1\le(n+1)^{1/\sqrt n}\le(2n)^{1/\sqrt n}=2^{1/\sqrt n}((\sqrt n)^{1/\sqrt n})^2$$

If we take $2^{1/x}$ and $x^{1/x}\to1$ as $x\to\infty$ for granted, then

$$2^{1/\sqrt n}((\sqrt n)^{1/\sqrt n})^2\to1\cdot1^2=1$$

and the Squeeze Theorem does the rest.