Limits form the foundation of calculus — they’re the bridge between algebraic thinking and the concepts of derivatives and integrals. On the AP Calculus AB & BC exams, limits appear in both multiple-choice and free-response questions, often as a starting point for bigger problems.
In this RevisionDojo guide, you’ll learn:
- The definition and meaning of limits
- Algebraic and graphical approaches
- Essential limit laws and formulas
- Special cases, including L’Hôpital’s Rule
- Common mistakes to avoid on the AP exam
📚 What Is a Limit?
The limit of a function describes the value that f(x)f(x) approaches as xx approaches a particular number (or infinity).
Example:
limx→2(3x+1)=7\lim_{x \to 2} (3x+1) = 7
As xx gets closer to 2, f(x)f(x) gets closer to 7.
🔍 Methods for Finding Limits
1. Direct Substitution
If the function is continuous at x=ax = a, simply plug in aa.
Example:
limx→4(x2−5)=11\lim_{x \to 4} (x^2 - 5) = 11
2. Factoring and Canceling
If substitution gives 0/00/0, factor and simplify.
Example:
limx→3x2−9x−3=limx→3(x−3)(x+3)x−3=6\lim_{x \to 3} \frac{x^2 - 9}{x - 3} = \lim_{x \to 3} \frac{(x - 3)(x + 3)}{x - 3} = 6
3. Rationalizing
Used for radicals. Multiply by the conjugate to remove the root.
4. Special Trig Limits
You must memorize:
limx→0sinxx=1andlimx→01−cosxx=0\lim_{x \to 0} \frac{\sin x}{x} = 1 \quad\text{and}\quad \lim_{x \to 0} \frac{1 - \cos x}{x} = 0
5. L’Hôpital’s Rule
If substitution yields 0/00/0 or ∞/∞\infty/\infty:
limx→af(x)g(x)=limx→af′(x)g′(x)\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}
Example:
limx→0sinxx=limx→0cosx1=1\lim_{x \to 0} \frac{\sin x}{x} = \lim_{x \to 0} \frac{\cos x}{1} = 1
6. Limits at Infinity
Focus on highest-degree terms for rational functions.
Example:
limx→∞3x2+15x2−4=35\lim_{x \to \infty} \frac{3x^2 + 1}{5x^2 - 4} = \frac{3}{5}
⚠️ Common AP Calculus Limit Mistakes
- Forgetting to check for continuity before applying methods
- Misapplying L’Hôpital’s Rule when it’s not an indeterminate form
- Ignoring units/context in applied problems
- Dropping negative signs during factoring
📊 Practice Strategy from RevisionDojo
- Drill basic limits daily until direct substitution and factoring are second nature
- Practice past FRQs with real-world context (rates, motion, area) involving limits
- Include at least 5 “trick” limits each week to prepare for curveballs on the exam
🧭 Final Advice from RevisionDojo
Limits are the gateway to derivatives — and the AP exam loves testing them in subtle ways.
The more methods you master, the faster you can decide which tool to use under pressure.
