What is the best way to solve a limit of $\log(x)^{\log(x)}$?

Question

I have this function which I need to calculate the limit of $\log(x)^{\log(x)}$ as $x$ approaches 1. While my mind tells me that with logic, $0+^0+$ equals to number to the power of 0, therefore = 1, but I don't know how to prove it.

Answer 1

Let $y = \lim_{x\to1} (\ln x)^{\ln x}$. Then we have $$\ln y = \lim_{x\to1}\frac{\ln(\ln x)}{1/\ln x} = \lim_{x\to1} \frac{\frac{1}{x \ln x}}{-\frac{1}{x(\ln x)^2}} = - \lim_{x\to 1} \ln x = 0$$ So $y = 1$.