Limits & Continuity

Limits are the foundation of AP Calculus AB — derivatives and integrals are both defined through them. A limit describes the value a function approaches as x approaches some point.

Evaluating limits

1. Direct substitution first. If lim(x→2) (x² + 1) you just get 5.
2. If substitution gives 0/0 (indeterminate), factor or simplify:

lim(x→3) (x² − 9)/(x − 3) = lim (x+3)(x−3)/(x−3) = lim (x + 3) = 6

One-sided limits & continuity

• lim(x→a⁻) approaches from the left, lim(x→a⁺) from the right.
• A function is continuous at a if the left limit, right limit and f(a) all exist and are equal.

Limits at infinity

Compare the highest powers. lim(x→∞) (3x² + 1)/(x² − 5) = 3 (ratio of leading coefficients).

Worked example

lim(x→0) (sin x)/x. This is a standard result: = 1 (know it by heart; it underpins the derivative of sin x).

AP strategy

When direct substitution gives 0/0, the question is testing algebra (factor, rationalise, or use a known limit) — not L'Hôpital, which is not on the AB exam. On the calculator-allowed section you can confirm a limit by evaluating the function very close to the point.