The real idea is much simpler than "what's the biggest impossible number", it's "what number am I getting closer to."
Also, there wasn't even an example of when a limit doesn't exist, that's quite a significant omission. The function f(x) = sin(1/x) is a good contrast to the previous function; in this case, the limit does not exist as x approaches 0, and it's easy to see here that it's because f(x) doesn't get closer to any value, it keeps oscillating wildly between -1 and 1.
So I would like to thank you once again for letting me know about this huge mistake of mine that I failed to notice. I was busy and didn't see your comment until now. I have just added clarification for that in the article.
The reason why there's so little about limit in this article was that I intended to go through the limit section as quick as possible and get started on differentiation and integration, which are what most readers are on my page for. On a second thought I decided to at least pen down the (ε, δ)-definition, and so I added the blockquote with the "biggest/smallest-impossible" analogy which unfortunately ended being the last thing you saw in my article before you closed your tab. I'm actually planning on writing another article, one specially for the concept of limit, covering topics from continuity, existence of limit (as you mentioned) and one-sided limit to sequence and taylor series to limit point, neighborhood and a bit of topology.
The reason limits are introduced in calculus is you need a Continuous function for the basic assumptions that allow integration/differentiation to work. Basically, the limits for all points you care about need to agree or you can't take the derivative. In other words f(x) ~= f(x + 1/ infinity) ~= f(x - 1/ infinity) for all x.
PS: I still like the overall presentation.
Other things that you would find in a lot of calculus textbooks but I didn't cover are complex function, the application of calculus (e.g. in Newtonian physics, in optimization problems) and stuff like L'Hospital's Rule, Squeeze theorem, etc. I didn't want to get into too much of the details in this article because my plan was to be concise and get straight to the points. I want to write it in a way that anyone who is new to calculus can grasp the concepts and have an idea of what calculus is about in the shortest time possible. On a side note, most of the functions they will be dealing with are elementary functions, which are continuous over their domains.