What you'll learn
Limits and Continuity is the foundation of AP Calculus AB (~10–12% of the exam). Limits underpin both the derivative and the integral, so master them early.
Evaluating limits
- Direct substitution first: lim(x→2)(3x + 1) = 7.
- If you get 0/0, factor or simplify: lim(x→3)(x²−9)/(x−3) = lim(x+3) = 6.
- Standard limit: lim(x→0)(sin x)/x = 1.
One-sided limits
lim(x→a⁻) and lim(x→a⁺) approach from left and right. The two-sided limit exists only if both one-sided limits are equal. If they differ, the limit does not exist (a jump).
Limits at infinity (end behaviour)
- lim(x→∞)(1/x) = 0.
- For rational functions, compare degrees: same degree → ratio of leading coefficients (e.g. (2x²+3)/(x²−1) → 2); numerator smaller → 0; numerator larger → ±∞.
Asymptotes
- Vertical asymptote: denominator → 0 (and numerator ≠ 0), e.g. 1/(x−5) at x = 5.
- Horizontal asymptote: from the limit at infinity.
Continuity
A function f is continuous at x = a if: f(a) exists, the limit exists, and lim(x→a) f(x) = f(a). Types of discontinuity: removable (hole), jump, and infinite (asymptote).
Intermediate Value Theorem (IVT)
If f is continuous on [a, b] and N is between f(a) and f(b), then f takes the value N somewhere in (a, b). Used to show a root exists.
Exam tips
- Always try substitution first; reach for factoring on 0/0.
- State the three-part continuity definition precisely.
- Cite "continuous on a closed interval" when applying the IVT.
Common mistakes
- Saying a limit exists when the one-sided limits differ.
- Forgetting that a removable discontinuity still means the function isn't continuous there.
- Confusing vertical and horizontal asymptote conditions.