Left Hand and Right Hand Limits: One-Sided Limits Guide

Some functions behave differently on the two sides of a point. A left-hand limit looks only at x values slightly smaller than a, written lim(x → a⁻) f(x). A right-hand limit looks only at x values slightly larger than a, written lim(x → a⁺) f(x). These are called one-sided limits, and they let us describe jumps, corners, and boundary behavior precisely.

One-sided limits matter because many real situations change their rule at a threshold. Tax rates, shipping charges, and speed limits all switch formulas at a cutoff point. In mathematics, piecewise-defined functions have exactly this character, and one-sided limits are the tool that reveals what happens on each side of the switch.

Key Concepts

• Left-hand limit: lim(x → a⁻) f(x). The minus sign means x approaches a from below (from the left on the number line).
• Right-hand limit: lim(x → a⁺) f(x). The plus sign means x approaches a from above (from the right).
• Existence rule: the ordinary two-sided limit lim(x → a) f(x) exists exactly when both one-sided limits exist and are equal to the same number L.
• Piecewise functions: functions defined by different formulas on different intervals are the most common place where one-sided limits disagree.

Formulas

• lim(x → a) f(x) = L if and only if lim(x → a⁻) f(x) = L and lim(x → a⁺) f(x) = L.
• If lim(x → a⁻) f(x) ≠ lim(x → a⁺) f(x), then lim(x → a) f(x) does not exist. We say the limit DNE.
• For f(x) = |x|/x: the right-hand limit at 0 is 1 and the left-hand limit at 0 is −1, so the two-sided limit does not exist.

Worked Example

Find the one-sided limits of f(x) = |x|/x at x = 0, and decide whether the two-sided limit exists.

For x greater than 0, |x| = x, so f(x) = x/x = 1. As x approaches 0 from the right, the function is constantly 1, so lim(x → 0⁺) f(x) = 1. Check numerically: f(0.1) = 1 and f(0.01) = 1. For x less than 0, |x| = −x, so f(x) = −x/x = −1. As x approaches 0 from the left, the function is constantly −1, so lim(x → 0⁻) f(x) = −1. Check: f(−0.1) = −1 and f(−0.01) = −1. The two one-sided limits are 1 and −1, which are not equal. Therefore lim(x → 0) |x|/x does not exist. The graph jumps from −1 to 1 at the origin.

Common Misconceptions

• If both one-sided limits exist, the limit exists. They must also be equal. Existence on each side is not enough by itself.
• A limit that DNE at one point means the function is broken everywhere. Non-existence is local. The function |x|/x has perfectly good limits at every x except 0.
• The minus and plus signs mean negative and positive x. They indicate the direction of approach relative to a, not the sign of x itself.

Real-World Applications

• Tax brackets: the marginal rate jumps at income thresholds, exactly like a piecewise function.
• Shipping and pricing tiers: cost per unit changes abruptly at quantity cutoffs.
• Digital signals: a switch flipping from off to on is a jump discontinuity, analyzed with one-sided limits.
• Speed limits: the legal maximum changes discontinuously at a zone boundary.

To summarize, one-sided limits examine a function from the left and from the right separately. When both sides agree on the same value, the ordinary limit exists and equals that value. When they disagree, the function jumps, and no two-sided limit exists at that point.