# Evaluate a limit involving logarithm.

Evaluate the following limit.

$$\lim_{x\to 1^+} (ln\ x)^{ln\ x}$$

I don't understand a step in the solution. The step says :

$$\lim_{x\to 1^+} \frac{1/[(lnx)(x)]}{-1/[(lnx)^2x]}$$

$$- \lim_{x\to 1^+} ln\ x = 0$$

I don't get what happened here.If i substituted the $x = 1$, i will get indeterminate form.

One approach: rewrite the limit byt setting $\log x =t$ and log it: $$\lim_{t \to 0} t^t=\lim_{t \to 0} t \log t=\lim_{t \to 0}\frac{\log t}{\frac{1}{t}}=0$$ The last step is by L'Hospital's rule. Hence the original limit is $e^0=1$