When computing the limit of a function at a point it is sometimes necessary to do some algebraic simplification of the function before you can substitute in the value. What we are actually doing is replacing the function with another function which is equal to it but has a larger domain (i.e. the simplified function has the value x=a in its domain). We can think of this as filling in the hole on the original function.
The original function has a hole at (i.e. is not in the domain), this is illustrated in the diagram below. However, when we simplify the function and we obtain a function which is equivalent to , for not equal to , and is itself defined at . In other words, we remove the hole. This allows us to compute the limit of at by computing the corresponding value of at .
Move the slider in the diagram below to remove the hole.
(Refer to example 7(a) in section 2.3 of the the class notes.)