One-Sided Limits

What Are One-Sided Limits?

Sometimes we only care about what happens as $x$ approaches a value from one direction—either from the left or from the right.

Notation

Left-hand limit (approaching from the left): $\lim_{x \to a^-} f(x)$

Right-hand limit (approaching from the right): $\lim_{x \to a^+} f(x)$

The superscript minus ($-$) means "from the left" and plus ($+$) means "from the right."

The Connection to Two-Sided Limits

For a regular (two-sided) limit to exist: $\lim_{x \to a} f(x) = L$

Both one-sided limits must exist and equal $L$: $\lim_{x \to a^-} f(x) = L \quad \text{and} \quad \lim_{x \to a^+} f(x) = L$

When Are One-Sided Limits Useful?

1. Piecewise Functions: Different formulas on each side of a point
2. Absolute Value Functions: Behavior changes at certain points
3. Jump Discontinuities: Different values on each side
4. Square Roots: Only defined for non-negative values
5. Rational Functions: Vertical asymptotes

Example: Piecewise Function

$f(x) = \begin{cases} x^2 & \text{if } x < 2 \\ 3x - 1 & \text{if } x \geq 2 \end{cases}$

Find $\lim_{x \to 2} f(x)$:

Left-hand limit: $\lim_{x \to 2^-} f(x) = \lim_{x \to 2^-} x^2 = 4$

Right-hand limit: $\lim_{x \to 2^+} f(x) = \lim_{x \to 2^+} (3x-1) = 5$

Since $4 \neq 5$, the limit does not exist (DNE).

Key Insight

One-sided limits help us understand exactly where and why a limit might fail to exist!