One sided limits

Several of the illustrations in the previous section relied informally on the idea of approaching a point from just one side; this section makes that idea precise.

Definition 9.2 (left-hand limit). The left-hand limit of $f$ at $x_0$ equals $l_1$ — written $\lim\limits_{x\to x_0^-} f(x)=f(x_0^-)=l_1$ — if $f(x)$ can be made arbitrarily close to $l_1$ by taking $x$ sufficiently close to $x_0$ while staying less than $x_0$.

Definition 9.3 (right-hand limit). The right-hand limit of $f$ at $x_0$ equals $l_2$ — written $\lim\limits_{x\to x_0^+} f(x)=f(x_0^+)=l_2$ — if $f(x)$ can be made arbitrarily close to $l_2$ by taking $x$ sufficiently close to $x_0$ while staying greater than $x_0$.

So the notation "$x\to x_0^-$" restricts attention to $x<x_0$ only, and "$x\to x_0^+$" restricts attention to $x>x_0$ only. (Figs. 9.6–9.9 contrast several situations: both one-sided limits existing and agreeing; both existing but disagreeing; a function that is simply not defined on one side of $x_0$ at all.)

Putting the pieces together, the two-sided limit $\lim\limits_{x\to x_0} f(x)=L$ exists exactly when all three of the following hold simultaneously:

1. $\lim\limits_{x\to x_0^+} f(x)$ exists,
2. $\lim\limits_{x\to x_0^-} f(x)$ exists, and
3. the two one-sided limits are equal, both equal to $L$.

Equivalently, $\displaystyle \lim_{x\to x_0} f(x)=L \iff \lim_{x\to x_0^-} f(x)=L=\lim_{x\to x_0^+} f(x)$. If even one of these three fails — one side does not exist, or the two sides disagree — the two-sided limit does not exist. It is worth stating explicitly that the existence of the one-sided limits is a strictly weaker requirement than the existence of the (two-sided) limit: a function can perfectly well possess both one-sided limits without those two values matching, whereas a genuine two-sided limit forces them to match.

A convenient alternative way to compute one-sided limits uses a small positive increment $h>0$ and lets $h\to0$:

$f(x_0^-)=\lim_{h\to 0} f(x_0-h), \qquad f(x_0^+)=\lim_{h\to 0} f(x_0+h).$ …