Why is $\frac{1}{\ln(0)} = 0$?
I thought $ln(0)$ is undefined.
The context is, I am looking for discontinuities in a function, and I expected $x=0$ to be a discontinuity since $ln(0)$ is undefined. Here's the relevant portion of the function.
$$f(x)=\frac{1}{\ln|x|+4}$$
I'm thinking that since $ln(0)$ is undefined that $x=0$. Therefore there is a discontinuity. Desmos says $f(0)=0$.
$\ln0$ is indeed undefined . As for the function, $\frac{1}{\ln|x|+4}$ would also consequently be undefined at $x=0$, unless, as kingW3 says in the comments, you separately define it to be some value at 0.
I'm not clear on why you asked about $\frac{1}{\ln0}$, though, since that isn't what the original function is, and thus may not share discontinuities with it.
About the Desmos result, calculators tend to approximate values sometimes, so you might not want to trust them too much.