The Idea of a Limit

The whole of differential calculus rests on one idea: the limit. Informally, we ask: as $x$ gets closer and closer to some fixed value $a$ (without necessarily ever equalling $a$), what value does $f(x)$ get closer and closer to?

We write this as

and read it as 'the limit of $f(x)$ as $x$ tends to $a$ is $L$'. Crucially, this statement says nothing about the value of $f$ at $x=a$ — $f(a)$ might not even be defined — it only describes the behaviour of $f(x)$ for $x$ near $a$.

To make 'near $a$' precise, we distinguish approaching from the left and from the right:

• The left-hand limit, written $\lim_{x \to a^-} f(x)$, is the value $f(x)$ approaches as $x$ increases toward $a$ from values less than $a$.
• The right-hand limit, written $\lim_{x \to a^+} f(x)$, is the value $f(x)$ approaches as $x$ decreases toward $a$ from values greater than $a$.

The (two-sided) limit $\lim_{x\to a} f(x)$ exists if and only if the left-hand and right-hand limits both exist and are equal; their common value is then the limit. …