The limit

Definition

A function $f$f tends towards $L$L as $x$x tends towards $a$a if $f(x)$f(x) can be made as close to $L$L as one wishes, by taking $x$x close enough to $a$a.

The point $a$a itself is excluded. That is the whole reach of the notion: a limit describes a neighbourhood, not a value.

What the definition must forbid

"As close as one wishes" is not a mathematical sentence until one says who chooses, and in what order. The so-called $\varepsilon$ε–$\delta$δ formulation fixes both:

$\forall \varepsilon > 0,\ \exists \delta > 0 : 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon$∀ε>0, ∃δ>0:0<∣x−a∣<δ⟹∣f(x)−L∣<ε

The adversary first imposes a margin $\varepsilon$ε on the arrival; one must then produce a margin $\delta$δ on the departure that guarantees it. The order of the quantifiers is the statement: $\delta$δ depends on $\varepsilon$ε, never the reverse.

The case that justifies it all

Consider $f(x) = \dfrac{x^2 - 1}{x - 1}$f(x)=x−1x2−1​.

At $x = 1$x=1 the expression is meaningless: the denominator vanishes. Yet for every $x \neq 1$x=1 it simplifies:

$\frac{x^2 - 1}{x - 1} = \frac{(x-1)(x+1)}{x - 1} = x + 1$x−1x2−1​=x−1(x−1)(x+1)​=x+1

The function therefore equals $x + 1$x+1 everywhere except at 1, where it is undefined. Its limit at 1 is 2 — a value the function never attains.

Three behaviours

The last two have no limit. Diverging and oscillating are two distinct ways of failing: the first has a direction, the second has none.

Continuity follows from it

A function is continuous at $a$a when three conditions hold together: $f(a)$f(a) exists, the limit at $a$a exists, and the two coincide.

$\lim_{x \to a} f(x) = f(a)$x→alim​f(x)=f(a)

The function of the previous section fails the first: its limit exists, its value does not. It can be made continuous by setting $f(1) = 2$f(1)=2 — what is called a continuous extension.

Summary

A limit says what a function tends towards, independently of what it equals at the point.