The point of using that example was that the limit can exist without function being continuous. we can intuitively see the function is not continuous at x =1, but we are not interested at exactly that point. We are interested in how the values are approaching. I may be relying in the concept of continuity at other points than x=1, but I still dont know why should I make function continuous at x=1.
RE: So close, no matter how far: Concept of Limit