Why is $\frac{1}{\ln(0)} = 0$?

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 $$f(0)=0$$.

• The function $f(x)=\frac{1}{\ln(x)}$ is indeed discontinuous at $x=0$. Showing the original function and context of your question may help us to answer your actual question though. Sep 25, 2021 at 20:53
• Maybe it's talking about the limit of it when $x$ goes to $0$ from right.
• You can define $f(0)=0$ and it will be continuous, with that said $f$ isn't defined on $x=0$ and $\ln(0)$ is not a thing. Sep 25, 2021 at 21:05
• Desmos also thinks that $10^{100}+2-10^{100} = 0$ desmos.com/calculator/lgzmcvfqzp , so you have to be careful with any calculator that you use. Sep 26, 2021 at 13:40
$$\ln0$$ 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.