Limits - I need a simple, easy to understande explanation Subject: Limits
I know what to do when I get $\frac{0}{0}$ or $\frac{nonzero}{0}$
(factor, common denominator, expand, multiply by conjugate ...)
But can someone help me understand why? 
What does $\frac{0}{0}$ mean? What does $\frac{nonzero}{0}$ mean? How does it look on the graph? And why the way to approach it is all of what I wrote above?
Thanks
 A: This is a non-rigorous way to look at limits, but I found it helps.
Sometimes a function can be well behaved, i.e. its plot is smooth, doesn't jump around the place and it is continuous. However, sometimes there might be isolated points which cause division by zero, in the form $\frac{0}{0}$, as you mentioned. 
Assuming you understand why $\frac{0}{0}$ is undefined, it seems rather irritating that you have a function which is working so well, but then it has this nasty undefined point. 
The concept of a limit is to say, 
hey, let's look at the points near the point that is undefined. Using them, can we somehow find a good value such that if this undefined point was defined, this is the value it would take?
Not too sure if this is what you're looking for, but hope it helps.
A: Here $\frac{0}{0}$ means only that you have a function in the form $\frac{f(x)}{g(x)}$ and when you compute the limit for $x\to x_0$ both the function have limit 0, so you can't use the property $\lim \frac{f(x)}{g(x)}=\frac{\lim f(x)}{\lim g(x)}$ and you have to compute in another way. The same holds for $\frac{nonzero}{0}$. There isn't a look on the graph because that limit could be a number, or $\infty$ or non existent. To see how that approach works you can think about polynomials. If $P$ is a polynomial and $P(a)=0$ you know that $P$ has a factor $(x-a)$. In the same way you can think that both your function have a "factor" $(x-a)$ and if you eliminate it from numerator and denominator you can get an easier limit.
