Difficulty: Intermediate | Prerequisites: Strong algebra skills, familiarity with functions and graphs, basic trigonometry.
Limits and derivatives form the backbone of AP Calculus BC. Everything that follows in the course, from integration to series, rests on whether you can evaluate a limit and take a derivative fluently. Limits formalise the idea of "what value does a function approach?", while derivatives answer "how fast is it changing right now?" If you are coming in cold, make sure you are comfortable with polynomial, rational, trigonometric, exponential, and logarithmic functions before going further.
A limit describes the value a function approaches as the input gets close to some number. A derivative measures the instantaneous rate of change of a function at a point, defined as a limit itself. Master the core differentiation rules (power, product, quotient, chain) and the major limit techniques (direct substitution, L'Hopital's Rule, degree comparison), and you have the tools for most of what Calculus BC throws at you.
Limit
The value that f(x) approaches as x approaches a given number a. In simple terms, it is the answer to "where is this function heading?"
Continuity
A function is continuous at a point if the limit exists there, the function is defined there, and the two values match. Think of it as being able to draw the graph through that point without lifting your pen.
L'Hopital's Rule
A technique for evaluating limits that produce an indeterminate form (0/0 or infinity/infinity) by differentiating the numerator and denominator separately. In simple terms, when plugging in gives you nonsense, take derivatives top and bottom and try again.
Removable discontinuity (hole)
A point where the limit exists but the function is either undefined or defined at a different value. Think of it as a single missing dot on an otherwise smooth curve.
Infinite discontinuity
A point where the function shoots off toward positive or negative infinity, creating a vertical asymptote.
Jump discontinuity
A break in a piecewise function where the left-hand and right-hand limits exist but are not equal. Common in absolute value and step functions.
Derivative
The instantaneous rate of change of a function at a point, equal to the slope of the tangent line there. In simple terms, the derivative tells you how fast something is changing right now.
Critical point
A value of x in the domain of f(x) where f'(x) = 0 or f'(x) is undefined. These are the candidates for local maxima and minima.
Mean Value Theorem (MVT)
If f is continuous on [a, b] and differentiable on (a, b), there is at least one point c where the instantaneous rate of change equals the average rate of change over the interval. Think of it as: somewhere on your drive, your speedometer matched your average speed.
Rolle's Theorem
A special case of the MVT where f(a) = f(b), guaranteeing a point c where f'(c) = 0. In simple terms, if you start and end at the same height, the curve must level out somewhere in between.
lim (x->a) f(x) = L means: as x gets arbitrarily close to a (from both sides), f(x) gets arbitrarily close to L. Start every limit problem by trying direct substitution first.
When direct substitution produces 0/0 or infinity/infinity, rewrite the expression as a fraction if it is not one already, then differentiate the numerator and denominator separately. Evaluate again. Repeat as many times as necessary until you get a determinate result.
Degree of numerator > degree of denominator: the limit is +/- infinity (sign depends on leading coefficients)
Degree of numerator < degree of denominator: the limit is 0
Degrees equal: the limit is the ratio of the leading coefficients
These same rules determine horizontal asymptotes.
Removable (hole): the limit exists, but the function value is missing or different. You find it by factoring and cancelling; the cancelled factor gives the x-coordinate of the hole.
Infinite (vertical asymptote): the function blows up. Set the denominator equal to zero and solve.
Jump: the left-hand and right-hand limits both exist but are not equal. Common in piecewise-defined and absolute value functions.
If f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists at least one c in (a, b) with f(c) = k.
The exam uses this to justify that a function must hit a particular value somewhere on an interval. You need continuity on the closed interval, and you need to show that k sits between f(a) and f(b).
f'(x) = lim (h->0) [f(x+h) - f(x)] / h
This is the formal definition. The exam may ask you to recognise a limit expression as a derivative in disguise.
Power Rule: f(x) = x^n, so f'(x) = nx^(n-1)
Product Rule: if f(x) = u * v, then f'(x) = u'v + uv'
Quotient Rule: if f(x) = u / v, then f'(x) = (u'v - uv') / v^2
Chain Rule: d/dx f(u) = f'(u) * du/dx. Always needed when differentiating a composite function.
d/dx sin(x) = cos(x)
d/dx cos(x) = -sin(x)
d/dx tan(x) = sec^2(x)
d/dx sec(x) = sec(x)tan(x)
d/dx csc(x) = -csc(x)cot(x)
d/dx cot(x) = -csc^2(x)
d/dx e^x = e^x
d/dx ln(x) = 1/x
d/dx a^(ux) = u * a^(ux) * ln(a)
If f is continuous on [a, b] and differentiable on (a, b), there exists some c in (a, b) such that:
f'(c) = [f(b) - f(a)] / (b - a)
The right-hand side is the average rate of change (the slope of the secant line). The theorem guarantees a point where the tangent line is parallel to it.
The special case where f(a) = f(b). The conclusion becomes f'(c) = 0 for some c in (a, b). In other words, if the function starts and ends at the same value, it must have a horizontal tangent somewhere in between.
A numerical technique for approximating solutions to differential equations. Given dy/dx = f(x, y) and an initial point (x_0, y_0):
x_new = x_old + delta_x
y_new = y_old + (dy/dx evaluated at the old point) * delta_x
Repeat for each step. The exam gives you a step size and a starting point and asks you to walk forward.
Category | Formula |
|---|---|
Limit definition | lim (x->a) f(x) = L |
L'Hopital's Rule | If 0/0 or inf/inf: differentiate top and bottom separately |
Derivative definition | lim (h->0) [f(x+h) - f(x)] / h |
Power Rule | d/dx x^n = nx^(n-1) |
Product Rule | d/dx (uv) = u'v + uv' |
Quotient Rule | d/dx (u/v) = (u'v - uv') / v^2 |
Chain Rule | d/dx f(u) = f'(u) * du/dx |
MVT | f'(c) = [f(b) - f(a)] / (b - a) |
Euler's Method | y_new = y_old + f'(x_old, y_old) * delta_x |
Students often think L'Hopital's Rule can be applied to any limit. It cannot. It works only when the limit produces an indeterminate form (0/0 or infinity/infinity). Applying it to a non-indeterminate form gives a wrong answer.
Students confuse the Product Rule with simply multiplying the two derivatives together. The correct form is u'v + uv', not u' * v'.
Students sometimes apply the Quotient Rule when simple algebra (cancelling a common factor) would resolve the limit. Always try simplification before reaching for heavier tools.
The derivative of cos(x) is -sin(x), not sin(x). The negative sign is easy to forget under exam pressure.
⚠️ The limit definition of the derivative appears on nearly every AP exam, often disguised as a limit expression you must recognise.
⚠️ The Intermediate Value Theorem is a favourite free-response justification. You must state that f is continuous on the closed interval and that the target value lies between f(a) and f(b).
⚠️ Mean Value Theorem questions frequently appear as: "Is there a value c such that..." You must verify continuity on [a, b] and differentiability on (a, b) before applying it.
⚠️ Euler's Method problems are typically 2-3 steps. Set up a table: x, y, dy/dx, delta_y. Work methodically; arithmetic errors are the main pitfall.
True or false: L'Hopital's Rule can be applied whenever a limit is difficult to evaluate.
Fill in the blank: d/dx tan(x) = ______
True or false: If f'(c) = 0, then f has a local maximum or minimum at c.
Fill in the blank: If degrees of numerator and denominator are equal, the limit as x approaches infinity is ______
True or false: The IVT requires f to be continuous on an open interval.
Answers: 1. False (only for indeterminate forms). 2. sec^2(x). 3. False (it could be an inflection point). 4. The ratio of leading coefficients. 5. False (it requires a closed interval [a, b]).
Q: Evaluate lim (x->3) (x^2 - 9) / (x - 3).
A: Factor the numerator: (x+3)(x-3)/(x-3). Cancel (x-3). Direct substitution gives 3+3 = 6.
Q: Find f'(x) if f(x) = x^3 * sin(x).
A: Use the Product Rule. f'(x) = 3x^2 * sin(x) + x^3 * cos(x).
Q: Does the Mean Value Theorem apply to f(x) = |x| on [-1, 1]? Why or why not?
A: No. Although f is continuous on [-1, 1], it is not differentiable at x = 0 (the graph has a corner), so the MVT hypotheses are not satisfied.
Q: Use Euler's Method with step size 0.5 to approximate y(1) given dy/dx = x + y and y(0) = 1.
A: At (0, 1): dy/dx = 0 + 1 = 1, so y(0.5) = 1 + 1(0.5) = 1.5. At (0.5, 1.5): dy/dx = 0.5 + 1.5 = 2, so y(1) = 1.5 + 2(0.5) = 2.5.
Q: Identify the type of discontinuity for f(x) = (x^2 - 4) / (x - 2) at x = 2.
A: Removable discontinuity (hole). The factor (x - 2) cancels, and the limit at x = 2 is 4, but f(2) is undefined.
Limits are the foundation of integration: the definite integral is defined as the limit of Riemann sums. If limits feel shaky, integration will too.
Derivatives connect directly to curve sketching (first and second derivative tests) and optimisation problems, both covered in the next set of notes.
The Mean Value Theorem reappears in the proof of the Fundamental Theorem of Calculus. Understanding MVT deeply makes the FTC feel less like magic.
limit, continuity, L'Hopital's Rule, L'Hospital's Rule, indeterminate form, 0/0, infinity over infinity, removable discontinuity, hole, vertical asymptote, horizontal asymptote, jump discontinuity, IVT, Intermediate Value Theorem, derivative, differentiation, power rule, product rule, quotient rule, chain rule, trig derivatives, sin cos tan sec csc cot, exponential derivative, logarithmic derivative, Mean Value Theorem, MVT, Rolle's Theorem, Euler's Method, tangent line, instantaneous rate of change, average rate of change, critical point, AP Calculus BC