Intuitive definition of a limit

Definition — Intuitive limit

We say that $\lim_{x\to x_0} f(x) = L$ if, as $x$ tends towards $x_0$ (without reaching it), the values $f(x)$ approach $L$ arbitrarily closely.

Remark

The limit describes the tendential behaviour of a function near a point (or at infinity), not the value of the function at that point. It is possible that $f(x_0)$ does not exist, or that it exists but differs from $L$: the limit “looks” only at what happens around $x_0$, not at $x_0$.

This distinction is the heart of the concept: precisely because the limit ignores the value at $x_0$ (and uses the punctured neighbourhood), it can describe the behaviour of functions that at $x_0$ are not even defined.

Links

Topics: Limits
Concepts: Limit