Left-hand and Right-hand (One-sided) Limits

$x$ can approach $a$ from two directions, and the function may behave differently on each side. This gives two one-sided limits:

• The left-hand limit (LHL), written $\displaystyle\lim_{x \to a^-} f(x)$, is the value $f(x)$ approaches as $x$ increases towards $a$ through values less than $a$.
• The right-hand limit (RHL), written $\displaystyle\lim_{x \to a^+} f(x)$, is the value $f(x)$ approaches as $x$ decreases towards $a$ through values greater than $a$.

The existence rule. The (two-sided) limit exists if and only if both one-sided limits exist and are equal:

If the two one-sided limits are different, the limit does not exist (DNE).

Illustration — limit exists. For $f(x) = x^2$ near $x = 3$: approaching from the left ($x = 2.9, 2.99, \dots$) gives values near $9$, and from the right ($x = 3.1, 3.01, \dots$) also gives values near $9$. LHL $=$ RHL $= 9$, so $\lim_{x\to 3} x^2 = 9$.

Illustration — limit does not exist. For the modulus function's slope-type ratio $f(x) = \dfrac{|x|}{x}$ near $x=0$: for $x>0$, $|x|=x$ so $f(x)=1$, giving RHL $=1$; for $x<0$, $|x|=-x$ so $f(x)=-1$, giving LHL $=-1$. Since $-1 \neq 1$, $\lim_{x\to 0} \dfrac{|x|}{x}$ does not exist.