-1

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 $f(0)=0$.

stefan-png
  • 29
  • 3
  • 4
    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. – ndhanson3 Sep 25 '21 at 20:53
  • 6
    Maybe it's talking about the limit of it when $x$ goes to $0$ from right. – Emad Sep 25 '21 at 20:54
  • 6
    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. – kingW3 Sep 25 '21 at 21:05
  • 2
    Desmos also thinks that $10^{100}+2-10^{100} = 0$ https://www.desmos.com/calculator/lgzmcvfqzp , so you have to be careful with any calculator that you use. – TomKern Sep 26 '21 at 13:40

1 Answers1

1

$\ln0$ 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.

harry
  • 1,076