Is $\frac{0}{\infty}$ indeterminate?

Question

I have been searching for an answer for this for half an hour and I can't seem to find one. I've lots of information about other combinations of $0$ and $\infty$ but I haven't seen anything that says whether $\frac{0}{\infty}$ is indeterminate?

Answer 1

This quotient is zero. This is true whether you mean projective infinity or extended positive infinity.

Since you say "indeterminate", I assume you're talking about limit forms. Well, division is continuous and defined at $0/\infty$, so this is, indeed, not an indeterminate form.

This is actually pretty easy to show that this limit form works out to $0$ by the $\epsilon-\delta$ definition of limit: you should try to do so on your own. (and ask a new question here on MSE about it if you get stuck)

Answer 2

It is better to rephrase the question like this:

If $\lim_{x \to a}f(x) = 0$ and $g(x) \to \infty$ as $x \to a$ then does limit $\lim_{x \to a}f(x)/g(x)$ exist?

The answer is yes it does and $\lim_{x \to a}f(x)/g(x) = 0$. It is best not to think of indeterminate forms as numbers and writing undefined expressions like $0/0, \infty/\infty, 0/\infty,\infty/0, \text{etc..}$ is a source of major confusion (although almost all the calculus text write in such notation and are guilty for spreading confusion). It is always better to state explicitly what one means by these forms by writing them in form of a limit.