Is $\frac{0}{0}$ different from $\frac{1}{0}$?

Question

In my mind, zero divided by zero answers the question of what $a$, when multiplied with zero, equals zero:

$a * 0 = 0$

Obviously, any real number will satisfy this equation. However, one divided by zero is different. It answers this question:

$a * 0 = 1$

This is different, because there are no solutions. Both results are referred to as "undefined". To me, these two "types" of undefined are completely different. I realize that many applications don't care whether there's no results or infinite results. However, am I correct in my assumption that there are two types of undefined here? And if so, is there any terminology differentiating them that I can search for, possibly related to sets?

Question summary: Are there different types of undefined? If so, what are they?

I'm sorry if this is a duplicate. I've searched, but it's kind of hard when you don't know what you're searching for. I have read this question, but it doesn't mention any specific terms.

Answer 1

The expression $\frac{0}{0}$ and, similarly, $\frac{1}{0}$ and anything divided by zero is called "undefined" for a reason. That's because there is no defined value for the expression; it's meaningless without context.

Let's look at the expression $\frac{0}{0}$ more closely in terms of limits. I ask you these following questions:

$1.$ What is $\lim \limits_{x \to 0} \frac{x^2}{x}$

$2.$ What is $\lim \limits_{x \to 0} \frac{x}{x}$

$3.$ What is $\lim \limits_{x \to 0} \frac{x}{x^2}$

All of these have an indeterminate form of $\frac{0}{0}$; however, each has a different limit. In this case, it is important to look at relative rates, i.e. how much "quicker" one becomes zero than the other.

Answer 2

In limits

$\frac{0}{0}$ is an indeterminate form. That is, if $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, then $\lim_{x \to a} \frac{f(x)}{g(x)}$ may or may not be finite. User92774 has given some examples.

However, $\frac{1}{0}$ is not an indeterminate form. If $\lim_{x \to a} f(x) = 1$ and $\lim_{x \to a} g(x) = 0$, then $\lim_{x \to a} \frac{f(x)}{g(x)} = \pm \infty$.

In computer floating-point arithmetic

In IEEE 754, 0.0 / 0.0 = NaN, but 1.0 / 0.0 = $\infty$.