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A two-sided limit $\lim_{x \to a} f(x)$ is considered to exist if and only if both the left-hand limit $\lim_{x \to a^-} f(x)$ and the right-hand limit $\lim_{x \to a^+} f(x)$ exist and are equal to the same finite value. If they are unequal, the two-sided limit does not exist.

Conditions: The function is defined in a neighborhood around $a$ (excluding possibly $a$ itself).; We are evaluating the limit as $x$ approaches $a$.

A two-sided limit $\lim_{x \to a} f(x)$ is considered to exist if and only if both the left-hand limit $\lim_{x \to a^-} f(x)$ and the right-hand limit $\lim_{x \to a^+} f(x)$ exist and are equal to the same finite value. If they are unequal, the two-sided limit does not exist.

Conditions: The function is defined in a neighborhood around $a$ (excluding possibly $a$ itself).; We are evaluating the limit as $x$ approaches $a$.