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AP Calc AB · Chapter 1 of 8

Limits and Continuity

Define limits numerically, graphically, and algebraically, and decide when a function is continuous at a point or over an interval.

Why this chapter matters

Limits are the entry point to calculus because both derivatives and integrals are defined using limiting behavior.

What you will learn

  • Estimate and evaluate one-sided and two-sided limits.
  • Determine continuity and classify removable, jump, and infinite discontinuities.
  • Use limit laws and algebraic simplification to compute limits.

Lessons in this chapter

  1. Reading limits from graphs and tablesInterpret approaching behavior without relying on direct substitution. Read the full guide →
  2. Limit laws and algebraic techniquesApply limit laws, factoring, and rationalization to resolve indeterminate forms.
  3. Continuity at a point and on an intervalCheck all continuity conditions and identify where they fail.
  4. Average and instantaneous change previewConnect secant slopes to the idea of a tangent slope via limits.

Study task

Given a piecewise function, find all points where continuity fails and justify each case with one-sided limits.

Chapter checkpoint

If f(x) = (x^2 - 1) / (x - 1) for x != 1, what is the limit of f(x) as x approaches 1, and is f continuous at x = 1?

Factor to f(x) = x + 1 for x != 1, so the limit is 2. The original function is not continuous at x = 1 unless f(1) is defined as 2.