Not as easy as it seems

We know that $\ln x > 1$ for $x>3$

So $x^{\ln x} > x^1=x$ for $x>3$

Since $\displaystyle \lim_{x\to\infty}x = \infty$ , we have $\lim_{x\to\infty}x^{\ln x} = \infty$

Both $x$ and $\ln(x)$ grow arbitrarily large as $x$ approaches $\infty$ , so the limit is $\boxed{\infty}$ .