One-sided limit

Definition

Let $D \subseteq \mathbb{R}$, let $f \colon D \to \mathbb{R}$, and let $a$ be a limit point of $D \cap (a, \infty)$. A real number $L$ is the right limit of $f$ at $a$ if the following holds. For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - L| < \varepsilon$ for all $x \in D$ with $a < x < a + \delta$. The notation is $\lim_{x \to a^+} f(x) = L$. The left limit $\lim_{x \to a^-} f(x)$ is defined in the same way, with $a - \delta < x < a$.

Remarks

The right limit is the limit of the restriction of $f$ to $D \cap (a, \infty)$. If $a$ is a limit point of points on both sides, then $\lim_{x \to a} f(x) = L$ if and only if both one-sided limits exist and equal $L$. For the sign function at $0$, the right limit is $1$ and the left limit is $-1$.

Explanations

L1 human

A one-sided limit only looks at points on one side of $a$. A function with a jump at $a$ has two different one-sided limits there.