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)?

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.
