Limits and continuity practice questions (AP Calculus)

Sample questions

All 2 free AP Calculus questions on this unit, from the free AP Calculus AB Practice Exam. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 12 marks · Limits
   Evaluate $\lim_{x \to 3} \dfrac{x^2 - 9}{x - 3}$.
   • A.
     $0$
   • B.
     $3$
   • C.
     $6$
   • D.
     The limit does not exist
   Show answerFrom the free AP Calculus AB Practice Exam →
2. Question 22 marks · Continuity & IVT
   A function $f$ is continuous on $[1, 4]$ with $f(1) = -2$ and $f(4) = 5$. Which of the following must be true?
   • A.
     f$ has a zero in the open interval $(1, 4)

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 2: f$̲ has a zero in …
   • B.
     f$ is increasing on $[1, 4]

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 2: f$̲ is increasing …
   • C.
     f'(c) = \frac{7}{3}$ for some $c$ in $(1, 4)

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 20: …) = \frac{7}{3}$̲ for some $c$ i…
   • D.
     f$ has exactly one zero in $(1, 4)

     LaTeX syntax issue: KaTeX parse error: Can't use function '$' in math mode at position 2: f$̲ has exactly on…
   Show answerFrom the free AP Calculus AB Practice Exam →

Key terms

The definitions and formulas AP Calculus expects for limits and continuity, typeset as they appear on the paper.

Limit

The value $f(x)$ approaches as $x$ approaches $a$ from both sides, whether or not $f(a)$ exists. The limit exists only when the left-hand and right-hand limits agree.

$$\lim_{x \to a} f(x) = L \iff \lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = L$$

Limit laws and algebraic techniques

Limits of sums, products and quotients follow the arithmetic, provided the limits exist and no division by zero occurs. For $\frac{0}{0}$ forms, factor and cancel, rationalise, or use a known limit such as $\lim_{x \to 0}\frac{\sin x}{x} = 1$.

Squeeze Theorem

If $g(x) \le f(x) \le h(x)$ near $a$ and $g$ and $h$ have the same limit $L$ at $a$, then $f$ has limit $L$ there too. Used for oscillating functions such as $x^2 \sin\frac{1}{x}$.

Continuity at a point

Three conditions: $f(a)$ exists, the limit at $a$ exists, and they are equal. Polynomials, exponentials, sines and cosines are continuous everywhere; rational functions wherever the denominator is non-zero.

$$\lim_{x \to a} f(x) = f(a)$$

Types of discontinuity

Removable (a hole: the limit exists but differs from $f(a)$ or $f(a)$ is undefined), jump (one-sided limits differ) and infinite (a vertical asymptote).

Limits at infinity and horizontal asymptotes

For rational functions compare the degrees of numerator and denominator: equal degrees give the ratio of leading coefficients, a smaller numerator degree gives 0, a larger one gives no horizontal asymptote.

$$y = L \text{ is a horizontal asymptote if } \lim_{x \to \pm\infty} f(x) = L$$

Intermediate Value Theorem (IVT)

If $f$ is continuous on $[a, b]$ and $k$ lies between $f(a)$ and $f(b)$, then $f(c) = k$ for some $c$ in $(a, b)$. On the exam, cite continuity explicitly when you use it to show a root exists.

Frequently asked questions

Other AP Calculus topics

• Differentiation: definition and basic rules
• Composite, implicit and inverse functions
• Contextual applications of differentiation
• Analytical applications of differentiation
• Integration and accumulation of change
• Differential equations
• Applications of integration
• Parametric, polar and vector functions (BC)
• Infinite sequences and series (BC)

All AP Calculus topics · AP Calculus exam guide · All free practice exams