A rigorous definition of the limit

Why formalise the limit?

At secondary school level, we say that “f(x) tends towards L as x tends towards a” without further clarification. At university level, a rigorous definition is required to prove theorems and avoid errors in intuitive reasoning.

Epsilon-delta definition

We say that lim(x→a) f(x) = L if: for any epsilon > 0, there exists a delta > 0 such that for any x, 0 < |x - a| < delta => |f(x) - L| < epsilon.

In other words, we can make f(x) as close to L as we like, provided we choose x sufficiently close to a (without being equal to a).

Limits at infinity

lim(x → +∞) f(x) = L means: for any ε > 0, there exists A such that x > A implies |f(x) − L| < ε. And lim(x → a) f(x) = +∞ means: for any M, there exists a delta > 0 such that 0 < |x - a| < delta => f(x) > M.

Left-hand and right-hand limits

A limit exists at a if and only if the left-hand limit (x → a-) and the right-hand limit (x → a+) exist and are equal. Classic example: f(x) = |x|/x has no limit at 0 because lim(x → 0-) f(x) = -1 and lim(x → 0+) f(x) = 1.

Common pitfall

Confusing ‘f(a) exists’ with ‘lim(x→a) f(x) exists’: a function may have a limit at a without being defined there (e.g. sin(x)/x at 0), or be defined at a without having a limit at that point.