I think I understand where you are coming from with the "without getting there" language because I used to actualky use that in teaching limits, although I sympathize with opponents of it.
Take an example of a limit if a sequence, (which is a function defined on the positive integers), like f(n) = 1/n. We ask for "the limit of f(n) as n goes to infinity". To me, the entries in the sequence are approximating values, and the limit is the number they are trying to approximate. So ask yourself what number do the numbers 1/n get closer to as n gets larger and larger. The answer is zero. That is, if you go out far enough in the sequence, the elements get, and remain, as close to zero as you might wish. I.e. if you want them to be closer than .0001, then after n gets larger than 10,000, every later sequence entry 1/n will be closer to zero than .0001. So zero is the limit of the sequence. In this case no element of the sequence ever equals zero, so although no entry in the sequence actually gets to zero, every element, far enough out, gets as close as desired.
Now here is another sequence: f(1) =1, f(2) = 2, f(3) = 3, f(4) = 4, f(5)= 1/5, f(6) = 1/6, f(7) =0, f(8) = 0, f(9) = 0, ...f(n) = 0 for every n ≥ 9. This sequence fools around for a while getting larger, but soon turns around and starts getting smaller, and all of a sudden jumps to zero, where it remains forever after. Since the only thing that matters in a sequence is where it eventually goes, this sequence is also (eventually) approximating to, i.e. heding ultimtely toward, zero. Indeed if you wnt to get an approximation to within .000001, of zero, you can take any sequence entry further out than f(7). So here again the limit is zero. But in this case, most elements of the sequence do actually get to zero, the limit.
So sometimes the sequence actualy gets to the limit "early", and sometimes it does not. This doies not matter at all for the notion of a limit. The limit is the number that the sequence ultimately zeroes in on, whether it actually gets there earlier or not does not matter.
I have to go now, but the case of functions is similar. I.e. when looking at the limit of a function like f(x) = (x^2-4)/(x-2), as x goes to 2, we are interested in what number is being approximated by these values f(x) when x is very near 2, but not equal to 2. That is because the function is not defined at 2. This is analogous to the fact the sequence function above was not defined at n = infinity.
It is more confusing to consider functions like f(x) = x^2, and ask for the ,imit as goes to 2, because this function is defined at 2, i.e. f(2) = 4. Since the limit of this function as x goes to 2, is also 4, you might say that the function does get to its limit, namely the limit as x goes to 2, is whatever number x^2 is fgettoing closer to as x gets closer to 2, and this is 4. But you are supposed to compute the limit only by looking at numbers x with x ≠ 2, so really, in my opinion this function also does not "get to 4", at least not as long as x has not got to 2.
The link is that some functions are defined at the point that x is going to, i.e. this f(x) = x^2 is defined at x=2, and one can ask two separate questions: 1) is there a limit (and what is it), of x^2, as x goes to 2 without reaching it? 2) does that limit, equal the value of f(2)? If so, the function is a good function, called continuous.
got to go.