Evaluate the following limit.
$$ \lim_{x\to \infty} (\ln\ x)^{\frac{1}{x}} $$
What i have tried:
$$ \ln\left[\lim_{x\to \infty} (\ln\ x)^{\frac{1}{x}}\right] $$
$$ \lim_{x\to \infty} \ln(\ln x)^{\frac{1}{x}} $$
$$ \lim_{x\to \infty} \frac{\ln(\ln x)}{x} $$
So as $ x$ approaches infinity, the limit goes to 0. But the answer in the book is 1.
Best Answer
You took the natural log $\ln$ of the limit to evaluate it easier, but you forgot to undo the natural log. It is just like how if you were to add $1$ to the limit to make it easier to calculate, you would have to subtract off $1$ in the end.
In this case, to "undo" a natural log, you take $e$ to the power of something. So after you took the natural log you calculated the limit to be $0$; then $$ e^0 = 1 $$ is your answer.