One-Sided Limits

A one-sided limit is simply a limit where you only approach from one direction, either from the left or from the right. It's useful because the two directions don't always lead to the same place, like the price of a package jumping once it passes a weight cutoff. The little symbols just tell you which way you're walking: $x\to a^-$ (coming from the left, using numbers a bit less than $a$) and $x\to a^+$ (coming from the right, using numbers a bit more than $a$). The big formula with the $\iff$ just says the ordinary limit exists when both walks end at the same $L$. It's one simple idea dressed up in symbols.

When we estimated limits with a table, we always checked values on both sides of $a$. That was for a reason. One of the ways a limit can fail is a jump, where the left side heads one place and the right side heads another. One-sided limits let us talk about each side separately: $x \to a^-$ means coming in from the left, and $x \to a^+$ means coming in from the right.

Piecewise functions are where you'll see this most. Take

Coming from the left, $x$ is less than 1, so you use the top formula and the values head to $1$. Coming from the right, you use the bottom formula and they head to $3$. Each side has a perfectly good limit, but they don't agree, so there's no single number $f(x)$ is heading to. $\lim_{x\to1} f(x)$ doesn't exist, and the graph jumps.

That's the whole rule: the two-sided limit exists exactly when both one-sided limits exist and match. A few other places this comes up:

• $\dfrac{|x|}{x}$ is $-1$ for $x < 0$ and $1$ for $x > 0$, so it jumps at 0.
• $\sqrt{x}$ is only defined for $x \ge 0$, so at $x = 0$ only the right-hand limit makes sense.
• Step functions like the Heaviside function $H(t)$, which switches from 0 to 1 at $t = 0$. Engineers use it to model a switch being flipped.

Thermostats, tax brackets and digital signals all switch at a threshold. One-sided limits describe the value just before and just after the switch.