I think my confusion of limits of sequences came from the definition of 'precision' that I was used to. You can say something is precise to mean that it is, in a sense, not messy. For example, 3 precisely divides 9, and we can write $a^2-b^2$ precisely as $(a-b)(a+b)$.
When we talk about dynamics or suchlike, quite often we use Taylor's theorem to get an approximation, and then things become imprecise. We started to simplify things, ignore forces, assume point masses, or spherically symmetric bodies, etc. This seems like a rather messy way to go about things (although this is not intended to spark an argument about whether or not this is true, it is just an opinion!)
So when we talk about limits, I suppose it helps to work with a different definition of precise (or at least, it helped me considerably).
When we talk about 'real-world' mathematics, nothing is exact. Circles don't exist, lines aren't one dimensional, and so on. But, you tell me exactly what you mean by 'precise', and I can give you an answer that satisfies your definition. It's not as if I can give you an estimate; I could literally go on forever refining my answer to whichever standards you set me. (That is, given any $\varepsilon>0$ I can find a term that is at most that far away from what I define as the limit, and I can carry on forever as $\varepsilon$ gets smaller).
I am not trying to explain what a limit is here, because you obviously already know, but what I hope to put across is how I had to change my idea of what was 'messy maths' and what was 'neat maths', and how limits seemed to make more sense when thinking of them as part of applied maths than pure.
Hope that helps in some way!