Variable change theorem for limits, why is it important that $g(x)\neq b$ at some punctured neighborhood of $x=a$?

Question

So the theorem is as follows.

Suppose that $\,f(g(x))\,$ is defined in some punctured neighborhood of $\,x = a\,$, and that $\,g(x)\neq b\,$ there.
If $\;\lim\limits_{x\to a}g(x)=b\;$ and $\;\lim\limits_{t\to b}f(t)=L\;,\;$ then $\lim\limits_{x\to a}f(g(x))=L\,$.

I'm wondering what the meaning of $\,g(x)\neq b\,$ is. Can someone give an example of why this is important? I'm guessing it is possible to replace that part with that $\,g\,$ is continuous.

It is simply in general we don't assume that $f(t)$ is defined at $t=b$. It is unnecessary to define $\lim_{t\to b}f(t)$. — Riemann, Dec 30 '22 at 07:32

Angelo · Answer 1 · 2022-12-30T07:47:40.620

2

It is so important that $\;g(x)\neq b\;$ on some punctured neighborhood of $\,x=a\,$, because, if $\,g(x)=b\,$ on that punctured neighborhood, $\,\lim\limits_{x\to a}f(g(x))=f(b)\,$ that in general is different from $L$.
But if the function $\,f(t)\,$ were continuous at $\,t=b\,,\,$ it would not be necessary that $\;g(x)\neq b\;$ on some punctured neighborhood of $\,x=a\,.$

edited Dec 30 '22 at 07:47

answered Dec 30 '22 at 07:39

Angelo

12,328

can u show an example why it is important that $g(x)\neq b$ on that punctured neighborhood. – per persson Dec 30 '22 at 18:56

Variable change theorem for limits, why is it important that $g(x)\neq b$ at some punctured neighborhood of $x=a$?

1 Answers1