The limit of a function f(x) at a given point x=a is, essentially, the value one would expect the function f(x) to take on at x=a if one were going solely by the graph. For example, if given a graph which resembles the function f(x) = x-1, one might expect the function to take on the value f(x) = 0 at x=1. However, the function f(x) = (x-1)^2 /(x-1) would also be graphed like f(x) = x-1, but would be undefined at x=1.
In the case listed above, one would analyze the situation by examining the function's behavior in the graph for x-values slightly above and slightly below the desired point. For this case, suppose one examines the graph at the points x= 0, x = 0.5, x = 0.75, x = 1.25, x=1.5, x=2. Doing this, we determine that as x->1 from both the right and the left, f(x) -> 0. Thus, the two-sided limit of the function f(x) = (x-1)^2 /(x-1) at x=1 is 0, though f(1) itself is undefined (as it takes on the form 0/0)
