Is this limit valid/defined and if so, what is the value of it? I was wondering if the following limit is even defined/valid; if it makes any sense. If so, what is the value of it? If not, why is it not defined/valid?
Define $n$:
$$ab=n$$ for $ a\rightarrow \infty$ and $ b\rightarrow 0$
Sure, this might be a weird question due to its perhaps philosophical nature. Please keep in mind that I am not yet that good at maths, so I would greatly appreciate an "understandable" answer.
Edit: I have read some of your answers. Does the following clarification change anything?
Let $a,b\in$ R
There is no "relation" between the two variables, one cannot be expressed using the other (such as $a=1/b$)
To rephrase the question: If two variables approach infinity and zero respectively, what would their product be (if it can be determined)?

Rather than "predefining n", I would approach it like this: Because a and b are two separate variables, their limits can be taken separately.
\begin{align*}
\lim_{a\rightarrow\infty}\lim_{b\rightarrow0}ab = \lim_{a\rightarrow\infty}(\lim_{b\rightarrow0}ab) = \lim_{a\rightarrow\infty}(a\times0) = \lim_{a\rightarrow\infty}0 = 0
\end{align*}
On the other hand, if you flip the limit-order,
\begin{align*}
\lim_{b\rightarrow0}\lim_{a\rightarrow\infty}ab = \lim_{b\rightarrow0}(\lim_{a\rightarrow\infty}ab) = \lim_{b\rightarrow0}(\infty\times b) = \lim_{b\rightarrow 0}\infty = \infty
\end{align*}
When you change the order of the limits, they evaluate to different things. If there was such a thing as $\lim_{a,b \rightarrow \infty, 0}ab$ then its value should be constant, which shows that the limit cannot exist (in the sense that it's undefined).
