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#)
