When we have an infinite limit at a finite point, we can prove it using a method similar to the proofs you saw earlier. This specific applet deals with the case where the limit is .
In this case, instead of letting someone pick an , no matter how small, we allow someone to pick an , no matter how large. If we can find a such that any -value in the range yields a value , then we have proven the limit. In this case we have let and you can select by using the slider. The value for is then calculated for you.
A very similar idea can be used to deal with the case where the limit is .