real analysis - Difference b/w Undefined and Indeterminate quantities. Can two of them be equaled to each other?

While studying limits (continuity of functions to be specific), I encountered with two indeterminate forms for LHL and RHL. Both being $\color{blue}{\frac{1}{0}}$ .

As per what I've read over internet, $\color{Blue}{\frac{1}{0}}$ isn't an indeterminate form. It is undefined quantity, to be specific. My question lies straight in the direction of forehead here. How will you differ an $\color{blue}{\text{undefined quantity}}$ and $\color{Red}{\text{indeterminate quantity}}$? They both don't exist in the practical universe. Then why do we consider them as different to each other?

Also, how illogical will it be to equate two indeterminate quantities $\left(\color{red}{0^0} = \color{Red}{0^0}\right)$ or two undefined quantities $\left(\color{blue}{\frac{1}{0} = \frac{0}{0}}\right)$ ?

It may sound too illogical to ask this question over here. But, I suppose, it will be better for me to get cleared this misconception before it deepens.

Answer

Undefined means the operation doesn't make sense. I.e. $\frac{1}{0}$ is undefined because $0$ does not have a multiplicative inverse (no number so that $0\cdot a = 1$).

Indeterminate refers to a form (usually arising from limits). If you look at a limit like $\lim_{x\to 0}\frac{\sin(x)}{x}$ you might say the "form" is $\frac{0}{0}$. The reason why the "form" is indeterminate is that given the form you cannot say what the value of the limit is (or if it exists). In this case, the limit evaluates to $1$. But, I could give you another limit whose form is $\frac{0}{0}$ and the limit is not $1$, e.g. $\lim_{x\to 1} \frac{x-1}{x^{n}-1}=\frac{1}{n}$. There are a variety of forms that are indeterminate but they all have the common thread that given the form you don't know (a) if the limit exists and (b) what the limit is.

As for equating indeterminate "quantities" it is invalid because (as I mentioned) you don't know what number the quantity actually represents, so you can't equate them. It would be ok to say that two limits have the same form... but I would hesitate to write it with an $=$.