AP Calculus AB and BC glossary

Limit

Also called: Limit of a function

A limit is the single value a function approaches as its input approaches a given number. The limit describes where the function is heading, not where it lands, so a limit can exist at a point where the function is undefined.

\lim_{x \to a} f(x) = L

Reading $\lim_{x \to a} f(x) = L$ out loud: as $x$ gets arbitrarily close to $a$ from either side, $f(x)$ gets arbitrarily close to $L$ . The value $f(a)$ never enters into it. That is the whole point of a limit, and it is what makes limits the right tool for defining a derivative, where the interesting quantity is undefined at the exact point you care about.

For the limit to exist, both one-sided limits must exist and agree. If the function approaches 3 from the left and 5 from the right, no single value describes where it is heading, so the limit does not exist.

The mistake

Assuming $\lim_{x \to a} f(x)$ always equals $f(a)$ . That is true only when $f$ is continuous at $a$ , which is a stronger condition, not the definition.

Appears in: Unit 1: Limits and Continuity