How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$

Question

How to solve $\lim _{k\rightarrow 1}\dfrac {1+\ln k}{\left| \ln \left( \ln k\right) \right| }$ I stucked at the denominator.

Answer 1

as $k\to1$, $\ln k\to0$, $\ln \ln k\to-\infty$ and $|\ln \ln k|\to\infty$, but $1+\ln k\to1$, so limit is $(\sim\frac1{\infty})\to0$

Answer 2

Here are the steps \[ \lim_{k\to 1} \frac{1+\ln k}{|\ln(\ln k)|}=\frac{1+\ln 1}{|\ln(\ln 1)|}=\frac{1+0}{|\ln 0|}=\frac{1}{|-\infty|}=\frac{1}{\infty}=0 \]

Answer 3

In a strict sense, such a limit does not exist, since $\log x$ is not defined in a left neighbourhood of the origin, while $\log x<0$ for any $x\in(0,1)$. On the other hand, $$\lim_{x\to 1^+}\frac{1+\log x}{|\log\log x|}=\frac{1}{\lim_{x\to 0^+}|\log x|}=+\infty.$$