Evaluate a limit involving logarithm.

Question

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.

Answer 1

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 $e^0=1$