Study guide
Calculus begins with a question algebra cannot answer on its own: what happens to a function as its input creeps toward a value it may never actually reach? This chapter builds the limit concept from the ground up, connects it to continuity, and uses it to define the derivative — the single idea that everything else in AP Calculus AB grows out of. Think of this chapter as learning the grammar before you write the essay.
Evaluating Limits Algebraically and Graphically
A limit describes the value a function approaches as x gets arbitrarily close to some number a, written lim(x to a) f(x) = L. The key insight beginners miss is that a limit does not care what f(a) actually equals, or even whether f is defined at a at all — it only cares about the trend on both sides. When direct substitution gives a real number, that number is the limit and you are done. When substitution gives the indeterminate form 0/0, algebraic rewriting almost always saves you: factor and cancel common terms, rationalize a numerator or denominator by multiplying by a conjugate, or combine fractions over a common denominator. For example, lim(x to 3) of (x^2 - 9)/(x - 3) looks undefined at x = 3, but factoring the numerator as (x-3)(x+3) lets the (x-3) terms cancel, leaving lim(x to 3) of (x+3) = 6. Trigonometric limits often lean on two special results: lim(x to 0) of sin(x)/x = 1 and lim(x to 0) of (1 - cos(x))/x = 0. A limit like lim(x to 0) of sin(5x)/x can be rewritten as 5 times sin(5x)/(5x), which approaches 5 times 1 = 5. For end behavior, as x approaches positive or negative infinity, compare the highest-degree terms of numerator and denominator: if degrees match, the limit is the ratio of leading coefficients, as in lim(x to infinity) of (3x^2 + 2x)/(5x^2 - 1) = 3/5. If the graph is described rather than given algebraically, read the y-value the curve is heading toward from the left and from the right separately before deciding whether a two-sided limit exists.
Continuity and Piecewise Functions
A function f is continuous at x = a when three conditions all hold: f(a) is defined, lim(x to a) f(x) exists (meaning the left-hand and right-hand limits agree), and that limit equals f(a). If any condition fails, the function has a discontinuity at that point — a removable discontinuity (a hole) when the limit exists but does not match f(a), a jump discontinuity when the one-sided limits disagree, or an infinite discontinuity where the function shoots toward positive or negative infinity, typically at a vertical asymptote. Piecewise functions are the natural testing ground for this definition because the exam frequently asks you to find a parameter that forces two pieces to meet smoothly. Suppose f(x) = x^2 + 1 for x < 2 and f(x) = kx + 3 for x is greater than or equal to 2. The left-hand limit as x approaches 2 is 2^2 + 1 = 5. For continuity, the right piece must also equal 5 at x = 2, so 2k + 3 = 5, giving k = 1. Continuity on a closed interval [a, b] means the function is continuous at every interior point and one-sided continuous at the endpoints. This matters because the Intermediate Value Theorem (IVT) requires it: if f is continuous on [a, b] and L is any value between f(a) and f(b), then there exists at least one c in [a, b] with f(c) = L. A classic use is proving a root exists — if f(1) = -2 and f(2) = 5, continuity guarantees some c between 1 and 2 where f(c) = 0, even though IVT never tells you the exact value of c.
Differentiability Versus Continuity
Every differentiable function is continuous, but the reverse is not true — continuity is necessary for differentiability, not sufficient. A function fails to be differentiable at a point for any of three reasons even when it stays continuous there: a sharp corner (like the absolute value function at x = 0, where the slope jumps from -1 to 1), a vertical tangent line (where the slope becomes infinite, as with the cube root function at x = 0), or a cusp (where the one-sided slopes both go to infinity but with opposite signs). Graphically, if you cannot draw one unique tangent line at a point, the derivative does not exist there, no matter how unbroken the curve looks. When comparing two piecewise pieces at a junction point, differentiability requires matching not only the function values (continuity) but also the derivative values computed from each piece; a mismatch in slope produces a corner even when the pieces connect without a gap. On a multiple-choice or free-response item, always check continuity first, since a discontinuous function is automatically non-differentiable there, saving you the extra work of testing derivatives from each side.
The Limit Definition of the Derivative
The derivative of f at a point x is defined as f'(x) = lim(h to 0) of [f(x+h) - f(x)]/h, provided that limit exists. This expression, called a difference quotient, represents the slope of the secant line through the points (x, f(x)) and (x+h, f(x+h)); as h shrinks toward zero, the secant line rotates into the tangent line, and its slope becomes the instantaneous rate of change at x. An equivalent form, useful when you are given a specific point a, is f'(a) = lim(x to a) of [f(x) - f(a)]/(x - a). Working directly from this definition is a common early-exam task designed to check that you understand where derivative rules come from rather than only how to apply them. For f(x) = x^2, finding f'(2) from the definition means computing lim(h to 0) of [(2+h)^2 - 4]/h. Expanding gives (4 + 4h + h^2 - 4)/h = (4h + h^2)/h = 4 + h, and letting h approach 0 yields f'(2) = 4 — matching what the power rule would give instantly, but demonstrating the underlying logic. The same limit process, applied at a general point x instead of a fixed number, produces the full derivative function, which is why the definition is often called the first-principles derivative. Expect problems that present a limit expression in this exact difference-quotient form and ask you to identify which function and which point it represents, without ever mentioning the word derivative.
Key terms
- Limit
- — The value a function approaches as the input approaches a given number, independent of whether the function is defined at that number.
- One-sided limit
- — The value a function approaches as x approaches a from only the left or only the right side.
- Indeterminate form
- — An expression such as 0/0 produced by direct substitution that requires algebraic simplification before the limit can be evaluated.
- Removable discontinuity
- — A point where the limit of a function exists but does not equal the function's value there, appearing as a hole in the graph.
- Jump discontinuity
- — A point where the left-hand and right-hand limits both exist but are not equal to each other.
- Intermediate Value Theorem (IVT)
- — If a function is continuous on a closed interval [a, b], it takes on every value between f(a) and f(b) at least once within that interval.
- Differentiability
- — The property that a function has a well-defined derivative (a unique tangent line slope) at a given point.
- Difference quotient
- — The expression [f(x+h) - f(x)]/h, representing the slope of a secant line, whose limit as h approaches 0 defines the derivative.
- Secant line
- — A line connecting two points on a curve; its slope is the average rate of change between those points.
- Tangent line
- — The line that touches a curve at a single point and matches the curve's instantaneous slope there.
- End behavior
- — How a function behaves as x approaches positive or negative infinity, often described using limits at infinity.
Exam tips
- When you see 0/0 after substitution, immediately try factoring, conjugate multiplication, or common denominators before assuming the limit fails to exist.
- Always verify continuity at a point before testing differentiability there — a discontinuity automatically rules out a derivative.
- For end-behavior limits of rational functions, compare degrees of numerator and denominator rather than plugging in infinity directly.
- When a problem gives you a bare limit expression with h approaching 0, check whether it matches the difference-quotient pattern before reaching for algebra tricks.
- For IVT problems, state the three ingredients explicitly in a free response: continuity on the interval, the two endpoint values, and the target value L lying between them.