Ah, thanks for the clarification. Let's start with a generic form of a limit that we can work from together:\[\lim_{x \rightarrow y} f(x).\]When we ask, "What is the limit as x approaches y?", we know that the function f(x) is not required to have a value at y. Here is an example: \[\lim_{x \rightarrow 1}\frac{ x^2+9x-10 }{ x-1 }.\]This is basically the function x+10=y with a missing point at x=1. We know that this line has a limit as x approaches 1 because the line is continuous everywhere except at x=1. I believe you know this, but I am laying foundation.
Now, as we consider the limit as x approaches y, in my first example, we never actually compute the value of y in the equation we're taking the limit of. And for good reason, this value of y is not required to exist in the function. So what the professor is saying, what looks like 0/0 is really a value approaching 0. There are an infinite number of values occupying the space on this approach to 0. Those are the values we are considering. Numbers such as 0.000000001/0.00000000000001 for instance.
When we plug in x-x0, we never actually plug in x-x. x-x0 just gets smaller and smaller and smaller, approaching zero, infinitely.