I have read the section of my text a couple times and still don't understand, how can I tell whether or not a function has a limit just by looking at it's associated graph
Basically, a limit is what it sounds like. A function f(x) has a limit, say b, at x = a if f(x) become closer and closer to b as x becomes closer to a.
The value of f(x) at x = a is irrelevant to whether f has a limit at a. However, if f does have a limit b, then putting f(a) = b makes f continuous at a. So, if f cannot be made continuous at a, it does not have a limit.
Another way to visualize this is to say that you can zoom indefinitely at x = a and still fit the graph of f in some neighborhood of x = a inside your screen. If, at some point, the graph goes through the top or bottom of the screen regardless of how small you make the segment around x = a, the function does not have a limit at a.
One important addition to the above reply is:
when we say that x is close to a we understand that $\displaystyle x\ne a$.
So if $\displaystyle \displaystyle\lim _{x \to a} f(x) = b$ then we know for any $\displaystyle x \approx a\;\& \,x \ne a$ we must have $\displaystyle f(x)\approx b$.