How can a limit exist if the function isn't defined/doesn't exist at that point? I hope I make sense, but I'm just a little confused and don't understand how a limit can exist at a certain point even if the function itself doesn't (is discontinuous)?
Thank you :)
 A: The idea of a limit is to look at the function in the general neighbourhood of the point at interest, without looking at the point itself. Since you can define a function pretty much arbitrarily, you can do something like this:
$$f(x) = \begin{cases}x \mbox{ if } x \neq 13 \\
0 \mbox{ if } x = 13\end{cases}$$
So for this function, $f(3) = 3$, $f(10) = 10$, $f(12.999) = 12.999$, $f(13.0001) = 13.0001$, and $f(13) = 0$. If we calculate the limit of $f(x)$ as $x \rightarrow 13$, then we see the function get closer and closer to equalling 13:
$$\begin{eqnarray} f(12) & = & 12 \\
f(12.9) & = & 12.9 \\
f(12.99) & = & 12.99 \\
f(12.9999999) & = & 12.9999999\end{eqnarray}$$
and so forth. We say that we can always get as close as we want to our function equalling 13 by choosing a value of $x$ close enough to 13. Importantly, it doesn't matter that $f(13) \neq 13$. In fact, even if we didn't set a value for $f(13)$, we can still say that $\lim_{x \rightarrow 13}f(x) = 13$ as long as the function is defined in the neighbourhood of $x = 13$.
