Walking half the distance
You step halfway to the door, then half of what's left, then half again — you get endlessly closer but a limit is the door you're approaching, not a step you take. Calculus is built on describing exactly what value you're heading toward.
The idea of a limit
The limit of f(x) as x approaches a is the value f(x) heads toward as x gets arbitrarily close to a — whether or not f is defined at a.
Existence
A two-sided limit exists only when the left-hand and right-hand limits agree.
f(x) = (x² − 1)/(x − 1) lim(x→1) f(x) = 2, since it simplifies to x + 1.