A limit describes what a function approaches as the input gets close to a certain value. It's written as:

This reads: "The limit of $f(x)$ as $x$ approaches $a$ is $L$."

Formally, it means: for every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

This $\epsilon$-$\delta$ definition is the rigorous foundation of limits, though intuitive understanding comes first.

Limits let us talk about function behavior near a point, even if $f(a)$ is undefined. They are critical to defining continuity and derivatives.