Mathematics

The key insight is that a limit exists at a point when the function's behavior becomes predictable and consistent as you approach that point from both directions. Rather than checking what $f(a)$ actually equals, you're asking whether the left-hand and right-hand limits agree—do values cluster around the same number no matter which side you come from? If $\lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) = L$, then the limit exists and equals $L$, regardless of whether $f(a)$ is defined or what it equals.

The key insight is that a limit exists at a point when the function's behavior becomes predictable and consistent as you approach that point from both directions. Rather than checking what $f(a)$ actually equals, you're asking whether the left-hand and right-hand limits agree—do values cluster around the same number no matter which side you come from? If $\lim_{x\to a^-}f(x) = \lim_{x\to a^+}f(x) = L$, then the limit exists and equals $L$, regardless of whether $f(a)$ is defined or what it equals.