One Sided Limits

One-Sided Limits: Examining Behavior from Each Direction

When analyzing how a function behaves near a point, sometimes approaching from different directions matters. One-sided limits let us examine this precisely.

The left-hand limit, denoted $\lim_{x \to a^-} f(x)$, represents the value $f(x)$ approaches as $x$ gets arbitrarily close to $a$ while remaining less than $a$. The superscript minus indicates "from the left."

The right-hand limit, denoted $\lim_{x \to a^+} f(x)$, represents the value $f(x)$ approaches as $x$ approaches $a$ while remaining greater than $a$. The superscript plus indicates "from the right."

When One-Sided Limits Differ

Some functions, especially piecewise-defined ones, behave differently on either side of a point. This is where one-sided limits become essential.

Consider $f(x) = \begin{cases} x + 1 & x < 2 \\ x^2 - 1 & x \geq 2 \end{cases}$. Approaching $x = 2$ from the left using the first rule: $\lim_{x\to 2^-} f(x) = 2 + 1 = 3$. From the right using the second rule: $\lim_{x\to 2^+} f(x) = 4 - 1 = 3$. Both equal 3, so the two-sided limit exists.

Now consider $g(x) = \begin{cases} x^2 & x < 1 \\ 3 - x & x \geq 1 \end{cases}$. From the left: $\lim_{x\to 1^-} g(x) = 1$. From the right: $\lim_{x\to 1^+} g(x) = 2$. Since $1 \neq 2$, the two-sided limit does not exist.

The Fundamental Theorem of Two-Sided Limits

$\lim_{x\to a^-} f(x) = \lim_{x\to a^+} f(x) = L \quad \Longleftrightarrow \quad \lim_{x\to a} f(x) = L$

This equivalence is crucial: a two-sided limit exists if and only if both one-sided limits exist and are equal. When they disagree, the two-sided limit simply fails to exist—a fundamental way limits can fail.

Key insight: The actual value $f(a)$ is irrelevant. Limits depend only on the approaching behavior near $a$, not at $a$ itself.