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.

Explore the definition with the simulation: the function $f(x)=\dfrac{x^2-4}{x-2}$ is not defined at $x=2$, yet the limit as $x \to 2$ exists and equals $4$.

Drag the slider $x$ towards $2$: the values $f(x)$ approach $L=4$ even though $f(2)$ does not exist.

Links

Topics: Limits
Concepts: Limit