estimate
An estimate is a controlled claim about size, like saying ‘this suitcase weighs at most 20 kilos’ without weighing it to the gram. You do not pin the quantity exactly — you fence it in with a bound you can actually justify, and that fence is often all you need.
In analysis an estimate is an inequality bounding the size of a quantity, typically of the form |quantity| <= (something manageable). Upper estimates show a quantity is not too big (hence small when the bound is small); lower estimates show it is not too small. Producing the right estimate — neither too weak to conclude nor harder to prove than necessary — is the core craft of the subject; estimates are, quite literally, the working currency of analytic arguments.
Good estimates are reusable: the triangle inequality, the bound |sin x| <= 1, geometric-series bounds and the mean value inequality |f(b) - f(a)| <= M|b - a| recur everywhere. The honest caveat: an estimate is a one-directional statement. Knowing |a_n| <= 1/n tells you a_n -> 0, but a bound that merely says |a_n| <= 5 tells you nothing about convergence — the bound must shrink, or be tied to epsilon, to do real work.
To show (n + 1)/(n^2 + 1) -> 0, estimate (n + 1)/(n^2 + 1) <= (n + n)/n^2 = 2/n for n >= 1. Since 2/n is arbitrarily small, so is the original term. The crude bound 2/n does all the work — no exact value is ever computed.
Replace a messy term by a clean bound that you can drive to zero.