The ε-δ definition pins down what 'limit' means rigorously.
ε-δ
lim(x→a) f(x) = L means: for every ε > 0, there exists δ > 0 such that |x − a| < δ implies |f(x) − L| < ε. Plain English: you can make f(x) arbitrarily close to L by taking x close enough to a.