Difficulty: Foundational | Prerequisites: Algebra II, understanding of function behaviour and graphs
Limits underpin everything in calculus. They define both the derivative and the integral. Before you can determine whether a function is differentiable at a point, you first need to know whether it is continuous there, and continuity itself is defined through limits. The AP exam tests these ideas both algebraically (evaluating limits of rational expressions) and graphically (reading behaviour from a graph to determine continuity and differentiability). These questions tend to be conceptual, and they reward careful reasoning over calculation speed.
A function is continuous at a point if the limit exists, the function value exists, and the two are equal. Differentiability is a stricter condition: the function must also have a well-defined, non-infinite slope. A function can be continuous without being differentiable (corners, cusps, vertical tangents), but it cannot be differentiable without being continuous.
Limit
The value that f(x) approaches as x approaches a particular value c. In simple terms, it describes the behaviour of the function near a point, regardless of the function's value at that point.
One-sided limit
The limit as x approaches c from only the left (x → c⁻) or only the right (x → c⁺). The two-sided limit exists only if both one-sided limits exist and are equal.
Continuity at a point
A function f is continuous at x = c if three conditions hold: (1) f(c) is defined, (2) lim(x→c) f(x) exists, (3) lim(x→c) f(x) = f(c). Think of it as: you can draw through the point without lifting your pen.
Differentiability at a point
A function is differentiable at x = c if the derivative f'(c) exists (i.e. the limit of the difference quotient exists and is finite). Differentiability requires continuity, plus the absence of corners, cusps, or vertical tangent lines.
Horizontal asymptote
A line y = L that the graph of f approaches as x → ∞ or x → –∞. Formally, lim(x→∞) f(x) = L. It describes end behaviour, not behaviour near a specific x-value.
When both the numerator and denominator are polynomials, the limit as x → ∞ depends on the leading terms.
Same degree: limit = ratio of leading coefficients
Numerator degree higher: limit = ±∞
Denominator degree higher: limit = 0
Worked example (from Q20): lim(x→∞) (2x³ + 3x² – 4x + 2) / (4x³ – x² + 5x – 3)
Both are degree 3. The limit is the ratio of leading coefficients: 2/4 = 1/2.
The answer is C) 1/2.
A piecewise function is continuous at the breakpoint if the left-hand piece and the right-hand piece approach the same value, and the function is defined to equal that value at the breakpoint.
Worked example (from Q11): f(x) = { –x + 5, x ≤ 2 } and { –(1/2)x + 4, x > 2 }
f(2) = –2 + 5 = 3 (from the left piece, which applies at x = 2)
lim(x→2⁻) f(x) = –2 + 5 = 3
lim(x→2⁺) f(x) = –(1/2)(2) + 4 = –1 + 4 = 3
Both limits equal 3, and f(2) = 3, so f is continuous at x = 2. ✓
The limit exists. ✓
For differentiability: the slope from the left is –1, and the slope from the right is –1/2. These are different, so there is a corner at x = 2, meaning f is not differentiable there. ✗
The answer is D) I and II only (continuous and limit exists, but not differentiable).
A function is continuous but not differentiable at a point where there is a sharp corner, a cusp, or a vertical tangent line.
Worked example (from Q6): From the graph shown, the function has:
A discontinuity (jump) at x = b (open and closed circles at different heights)
A corner at x = d (two different slopes meeting)
At x = d, the function is continuous (no gap) but has a sharp change in direction, making it not differentiable. The answer is D) d.
Worked example (from Q14): If y = 3 is a horizontal asymptote of f for x ≥ 0, which must be true?
A horizontal asymptote means lim(x→∞) f(x) = 3. This is the definition. The function can still equal 3 at some finite x-values, and f(3) can be perfectly well-defined, so options A and B are not necessarily true. Only option C must hold.
The answer is C) lim(x→∞) f(x) = 3.
Worked example (from Q86): For which graphs does lim(x→3) f(x) exist?
The limit exists if the left-hand and right-hand limits are equal. It does not matter whether f(3) equals that value, or whether f(3) is defined at all.
Graph I: the function approaches the same y-value from both sides at x = 3 (even with an open circle). Limit exists.
Graph II: the function approaches different values from the left and right. Limit does not exist.
Graph III: both sides approach the same y-value. Limit exists.
The answer is E) I and III only.
Limit of a rational function at infinity (same degree):
lim(x→∞) (aₙxⁿ + ...) / (bₙxⁿ + ...) = aₙ/bₙ
Continuity checklist at x = c:
f(c) is defined
lim(x→c) f(x) exists (left limit = right limit)
lim(x→c) f(x) = f(c)
Differentiability requires continuity plus:
lim(h→0) [f(c+h) – f(c)] / h exists and is finite
Continuity ensures that physical systems behave predictably: a temperature function is continuous in space (no instant jumps), and a position function is continuous in time. Differentiability is a stronger requirement; it fails at moments of sudden change, such as a ball bouncing (velocity has a corner). Understanding these distinctions matters in modelling: if your model has a corner, the derivative does not exist there, and any analysis relying on the derivative needs a different approach at that point.
Students often believe that if a limit exists at a point, the function must be continuous there. The limit can exist while f(c) is undefined or has a different value (a removable discontinuity).
A common error is assuming that a function with a horizontal asymptote at y = 3 can never touch the line y = 3. The asymptote describes behaviour as x → ∞, not a boundary the function cannot cross.
Students sometimes confuse "continuous" and "differentiable." A corner in the graph is continuous (no break) but not differentiable (no single tangent line). These are different properties.
When reading limits from graphs, the value f(c) (whether a solid dot exists) is irrelevant to whether the limit exists. Only the behaviour approaching x = c matters.
⚠️ Piecewise continuity and differentiability questions appear every year. Check the value from each piece at the breakpoint, and compare the slopes.
⚠️ Limits at infinity of rational functions are among the fastest free marks on the exam. Know the degree-comparison shortcut.
⚠️ "Continuous but not differentiable" is a favourite exam phrase. Look for corners, cusps, or vertical tangents.
⚠️ Horizontal asymptote questions test definitions precisely. The asymptote is about x → ∞ behaviour, nothing else.
True or false: If lim(x→3) f(x) = 5 and f(3) = 5, then f is continuous at x = 3.
Fill in the blank: Differentiability at a point requires ______ at that point.
True or false: A function with a corner is discontinuous at that corner.
Fill in the blank: lim(x→∞) (5x² + 1) / (10x² – 3) = ______.
True or false: If y = 4 is a horizontal asymptote, then f(x) can never equal 4.
Answers: 1. True. 2. Continuity. 3. False; a corner is continuous but not differentiable. 4. 1/2. 5. False.
Q: f(x) = { –x + 5 for x ≤ 2, –(1/2)x + 4 for x > 2 }. Is f continuous at x = 2? Is it differentiable there?
A: f is continuous at x = 2 (both pieces give 3), and the limit exists. However, f is not differentiable because the slopes from the left (–1) and right (–1/2) differ.
Q: What is lim(x→∞) (2x³ + 3x² – 4x + 2) / (4x³ – x² + 5x – 3)?
A: 1/2. Both polynomials are degree 3, so the limit is the ratio of leading coefficients.
Q: If y = 3 is a horizontal asymptote of f for x ≥ 0, which must be true: f(3) is undefined, f(x) ≠ 3 for all x ≥ 0, or lim(x→∞) f(x) = 3?
A: lim(x→∞) f(x) = 3 is the only statement that must be true. The other claims do not follow from the definition of a horizontal asymptote.
Q: At which values of x does lim(x→3) f(x) exist if the graph approaches the same y-value from both sides? What if there is an open circle at x = 3?
A: The limit exists whenever the left and right approaches agree. An open circle at x = 3 means f(3) may be undefined, but the limit still exists.
Limits connect to derivatives (the derivative is defined as a limit of the difference quotient), integrals (the definite integral is a limit of Riemann sums), and L'Hôpital's Rule in BC Calculus. Continuity is also the precondition for the Intermediate Value Theorem, the Extreme Value Theorem, and the Mean Value Theorem, all of which appear on the AB exam.
limits, continuity, differentiability, piecewise functions, horizontal asymptote, one-sided limits, limit at infinity, removable discontinuity, jump discontinuity, corner, cusp, AP Calculus AB limits