Limits and Continuity Summary

Calculus 1 · Axiom Academy

Unit 1 recap — the limit idea, the techniques for evaluating limits, and what it means for a function to be continuous. A limit is the value a function approaches as the input nears a target — the function need never actually reach it. Direct substitution is always your first move; it works whenever the function is continuous at the point. An indeterminate form ( or ) is a signal to do more work — factor, rationalize, use a known limit, or apply L'Hôpital's Rule. Continuity at a point requires three things at once: the value exists, the limit exists, and the two are equal. Limits at infinity describe end behavior and pin down horizontal asymptotes; the IVT and the Squeeze Theorem are the two big existence tools. The limit captures what value f(x) approaches as x gets arbitrarily close to a — from either side, but never requiring x = a itself. The function may be undefined at a and still have a limit there. Key idea: the limit is about the journey toward a , not the value at a . Watch out for: existing does not mean f(a) exists. The left-hand limit ( ) and right-hand limit ( ) approach a from below and above. The two-sided limit exists only when both one-sided limits exist and agree . When to use: piecewise functions, jumps, and reading limits off a graph. Watch out for: if the sides disagree, the two-sided limit does not exist.

This is the written version of the interactive lesson above. See the full Calculus 1 course.