You can substitute continuity in my reply by approaching something. It is not necessary to define it formally.
In terms of a series or sequence your approach makes sense since they are discrete since they are evaluated for 1,2,3,4 etc. But limit in the sense of this function f that you used is in a sense different since you are relying on its continuity by approaching in a continuous sense a certain value.
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.
I think you misunderstand my previous comment.
If a function is defined in some interval then the concept of limit is a bit different because you can use different ways to approach the same point. While if it is only discretely defined you can only do this over the integers. So my claim is that it is better to keep it in a discrete setting if you only define the limit in a discrete setting