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)?
 A: There are many limits we run across that are of the form $0 \cdot \infty$.  This is known as an indeterminate form and needs closer study, which requires a clearer definition than you have supplied.  Each of $a$ and $b$ normally comes with a formula.  Often those depend on a parameter that is common to them.  For example, we might have $\lim_{c \to \infty} c^2(\frac 1c)$.  In that case $a=c^2 \to \infty$ and $b=\frac 1c \to 0$  The product goes to infinity.  Alternately, you could have $\lim_{c \to \infty} c(\frac 1{c^2})$ with the opposite behavior.  
With the update, you cannot sensibly define a limit of the product.  The limit would depend on how fast $a$ goes to $\infty$ compared to how fast $b$ goes to zero.  As you have not specified how the limits are taken, the product is only defined if $a$ and $b$ have separate limits that do not result in an inderminate form.
A: 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).
A: The limit is undefined because it isn't specified at what rate $a$ and $b$ are approaching $\infty$ and $0$ relative to each other.
For example, if you would define $b$ as $1\over a$,then as $a \rightarrow \infty$ and $b \rightarrow 0$, $n \rightarrow 1$
However, if you would define $b$ as $1\over a^2$, then as $a \rightarrow \infty$ and $b \rightarrow 0$, $n \rightarrow 0$
Since the relation between $a$ and $b$ is not given, the limit remains undefined.
If as you mention in the edit there is no relation between $a$ and $b$, then there would be no limit for $n$. If $a$ would grow faster proportionally to $b$ getting smaller, $n$ would get larger, and if $b$ would shrink faster, then $n$ would get smaller. Since $n$ is not closing in on a certain number (or $\pm \infty$), $n$ does not have a limit.
A: This is known as an indeterminate form. In short, whether it's defined or not, and in case it is what value it is, depends on how quickly $a$ grows and how quickly $b$ shrinks.
Example. Let $a = k$ and $b = c/k$ where $c$ is a constant. Then
$$ab = k(c/k) = c$$
for all $k$. Hence the limit as $k\to \infty$ is simply $c$. What this means is that the limit can equal any constant depending on the value of $c$.
