proving limit of sequence when related limit is known

Question

Suppose I have the sequence $\{x_n\}$, where $ x_n >0$ for all $n\ge1$.

The sequence satisfies:

$$\lim_{n\rightarrow\infty}{x_n^{x_n}} = 4$$

I am trying to prove that

$$\lim_{n\rightarrow\infty}{x_n} = 2$$ I have tried logarithms and the definition of the limit, but to no avail...

Any hints?

Answer 1

Hint: You could e.g. show and use that $f(x)=x^x$ is continuous and strictly increasing for $x\geq 1$ (and $f(x)\leq 1$ for $x\leq 1$).