$\frac{1}{\infty-\infty}$ is an indeterminate form?

Question

I know that $$\infty-\infty$$is an indeterminate form .. What about $$\frac{1}{\infty-\infty}$$ is it an indeterminate form or is it equal to zero ?

Answer 1

It is easy to construct examples where the limit is $0$ such as

$$\lim_{x\to\infty}\frac{1}{x^2-x}=0$$

But other examples can be constructed such as

$$\lim_{x\to\infty}\frac{1}{(x+a)-x}=\frac{1}{a}$$

So it is an indeterminate form.

Answer 2

Substituting variables in indefinite forms returns indefinite forms. This is easy to see because a variable always has a scope of values you can replace it with (e.g. real numbers when $x\in\mathbb {R} $) An indefinite form is never part of any set, hence never in the scope of any variable. Therefore substituting one with something undefined is undefined again.