The graph of log(x) is 
When you're graphing $\frac{1}{log(x)}$;
Why is it that the reciprocal of log|x| at x=0 is taken to be 0, I.e., why is it continuous there? Isn't it just a limiting value towards infinity? Conversely, on zooming into the second graph (I plotted it online) f(0 never really touches 0).
Why does that happen, too?
On
The log function $\log_e x \equiv \ln x$ is not defined at $x=0$. More precisely, it is only defined for $x >0$. You are correct when you're saying that this is a "limiting value".
Specifically, it is:
$$\lim_{x \to 0} \ln x = - \infty \implies \lim_{x \to 0} \frac{1}{\ln x} = 0.$$
The thickness of the plotted line does seem misleading, but the value $0$ does not belong to the function plotted - thus it should have a "hole" there.
It isn't taken to be 0.
At $x = 0$, neither $\log|x|$ nor $\frac{1}{\log|x|}$ exist.
However, the $\lim_{x \to 0} \frac{1}{\log|x|}$ does exist and is equal to $0.$
Addendum
Per user request
Absolutely. At $x=0$, neither $\log x$ nor $\frac{1}{\log x}$ exist. I think the thickness of the line in the graph is misleading.