Concept preview

What you will be able to do

1. 01Evaluate left-hand and right-hand limits from graphs, tables and piecewise formulas.
2. 02Determine whether a two-sided limit exists by comparing the two one-sided limits.
3. 03Analyse limits of the absolute-value, greatest-integer (floor) and piecewise-defined functions using one-sided reasoning.
4. 04Interpret one-sided limits at domain endpoints and at jump and infinite discontinuities.
5. 05Diagnose and repair a flawed one-sided-limit argument.

Until now a limit has asked what a function heads toward as $x$ closes in on a point. But a number can be approached from two sides, and sometimes the story differs depending on which side you come from. A one-sided limit records the behaviour from a single direction. The left-hand limit, written $\lim_{x\to c^-} f(x)$, uses only inputs slightly less than $c$; the right-hand limit, written $\lim_{x\to c^+} f(x)$, uses only inputs slightly greater than $c$. The little superscript minus or plus is not a sign on a number — it is an arrow telling you which side of $c$ you are walking in from.