Left-Hand Limit, Right-Hand Limit and Existence of a Limit

A value of $x$ can approach a point $a$ from two directions — from values less than $a$ (from the left) or from values greater than $a$ (from the right) — and a function's tendency can genuinely differ depending on which side it is approached from.

• The left-hand limit (LHL) of $f(x)$ as $x \to a$ is the value $f(x)$ approaches as $x$ approaches $a$ through values smaller than $a$, written $\lim_{x \to a^{-}} f(x)$
• The right-hand limit (RHL) of $f(x)$ as $x \to a$ is the value $f(x)$ approaches as $x$ approaches $a$ through values greater than $a$, written $\lim_{x \to a^{+}} f(x)$

The existence condition. The (two-sided) limit $\lim_{x \to a} f(x)$ exists if, and only if, both one-sided limits exist and are equal: $\lim_{x \to a^{-}} f(x) = \lim_{x \to a^{+}} f(x) = L$ If the LHL and RHL are unequal, the two-sided limit at $a$ does not exist, even though each one-sided limit individually might exist perfectly well on its own. …