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 $a - \delta < x < a$.