Left- and Right-Hand Limits

The left-hand limit $\lim_{x \to x_0^-} f(x)$ is the value $f(x)$ approaches as $x$ approaches $x_0$ from the left (i.e., $x < x_0$). The right-hand limit $\lim_{x \to x_0^+} f(x)$ is the value approached from the right ($x > x_0$).

The full limit $\lim_{x \to x_0} f(x)$ exists if and only if both one-sided limits exist and are equal. If the left-hand and right-hand limits differ, the overall limit does not exist. This happens for step functions and piecewise functions with jumps.

For example, the Heaviside step function $H(x) = 0$ for $x < 0$ and $H(x) = 1$ for $x \geq 0$ has $\lim_{x \to 0^-} H(x) = 0$ but $\lim_{x \to 0^+} H(x) = 1$. The overall limit $\lim_{x \to 0} H(x)$ does not exist.