IB Maths AA SL · Unit 5: Calculus
IB Maths AA SL Limits and Rates of Change Questions
Exam-style IB Maths AA SL limits and rates of change questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 16 questions
- Paper 1: 9
- Paper 2: 7
- 4 easy
- 5 medium
- 4 hard
- 3 very hard
- 3 worked examples
Practise Limits and Rates of Change questions →
AA SL formula booklet
What's examined in AA SL limits and rates of change
The question bank covers these limits and rates of change question types (number of questions in brackets):
- Derivative concepts & applications (9)
- Evaluating limits (4)
- Average rates of change (3)
Limits and Rates of Change worked examples
Worked example 1: Reading graphical limits and function values · easy
The graph of a piecewise function $y = f(x)$ is a straight line from $x = -1$ to $x = 5$, but has a discontinuous "hole" at $(3, 4)$ and an isolated solid point at $(3, 2)$. Evaluate $f(3)$ and $\displaystyle\lim_{x \to 3} f(x)$.
1. Understand: $f(3)$ asks for the strictly defined $y$-coordinate of the function at exactly $x = 3$.
2. Identify the solid dot at $x = 3$, which sits at $y = 2$. So $\mathbf{f(3) = 2}$.
3. Understand: $\lim_{x \to 3} f(x)$ asks for the value the graph is heading towards as $x$ approaches $3$ from both sides.
4. Trace from the left ($x \to 3^-$): the line heads for the hole at $y = 4$.
5. Trace from the right ($x \to 3^+$): the line also heads for the hole at $y = 4$.
6. Conclude: both one-sided limits agree, so $\mathbf{\lim_{x \to 3} f(x) = 4}$.
Examiner tip: A limit describes where the function is HEADING and ignores what happens exactly at that point. $f$ does not even need to be defined at $x = c$ for $\lim_{x \to c} f(x)$ to exist.
Worked example 2: Derivative from first principles · medium
The mathematical definition of the derivative is $f'(x) = \displaystyle\lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Use this formal limit definition (first principles) to prove that the gradient function of $f(x) = x^2$ is $2x$.
1. Substitute: $f'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}$.
2. Expand the numerator: $\lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h}$.
3. Cancel $x^2 - x^2$: $\lim_{h \to 0} \frac{2xh + h^2}{h}$.
4. Factorise the numerator: $\lim_{h \to 0} \frac{h(2x + h)}{h}$.
5. Cancel the common $h$: $\lim_{h \to 0} (2x + h)$.
6. Evaluate by direct substitution $h = 0$: $2x + 0 = \mathbf{2x}$.
Examiner tip: Write "$\lim_{h \to 0}$" on every line of working until the exact moment you substitute $h = 0$. Dropping the notation prematurely is a notation error.
Worked example 3: Limits at infinity and horizontal asymptotes · hard
Evaluate $\displaystyle\lim_{x \to \infty} \frac{3x + 1}{x - 2}$ and briefly explain the geometric significance of this limit regarding the graph of the rational function.
1. Strategy: divide every term by the highest power of $x$ in the denominator ($x^1$).
2. Divide: $\lim_{x \to \infty} \frac{\frac{3x}{x} + \frac{1}{x}}{\frac{x}{x} - \frac{2}{x}}$.
3. Simplify: $\lim_{x \to \infty} \frac{3 + \frac{1}{x}}{1 - \frac{2}{x}}$.
4. Evaluate: as $x \to \infty$, $\frac{1}{x} \to 0$ and $\frac{2}{x} \to 0$.
5. Substitute: $\frac{3 + 0}{1 - 0} = \mathbf{3}$.
6. Explain: the limit is $3$, which proves there is a $\mathbf{\text{horizontal asymptote at } y = 3}$.
Examiner tip: SL shortcut: for a rational function where the highest power of $x$ matches in numerator and denominator, the limit at infinity is just the ratio of their leading coefficients ($\frac{3}{1} = 3$).
Try these IB Maths AA SL limits and rates of change questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
The derivative of a function \(f(x)\) is known as its gradient function, denoted by \(f'(x)\).
Given that \(f(x) = -x^2 + 4x + 5\) and its derivative is \(f'(x) = -2x + 4\):
Find the exact gradient of the curve at \(x = 3\).
State whether the function \(f(x)\) is increasing or decreasing at \(x = 3\). Give a reason for your answer.
Question 2 · medium · 5 marks · Paper 1
The mathematical definition of the derivative \(f'(x)\) is given by the limit:
\[f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}\]
Let \(f(x) = x^2\).
Expand and simplify the expression \(\frac{f(x+h) - f(x)}{h}\).
Hence, evaluate the limit as \(h \to 0\) to find the gradient function \(f'(x)\).
Question 3 · hard · 5 marks · Paper 1
Consider the rational function \(f(x) = \frac{3x + 1}{x - 2}\), where \(x \neq 2\).
By considering the behaviour of the function as \(x\) becomes very large, evaluate \(\lim_{x \to \infty} f(x)\).
State the geometric feature of the graph of \(f(x)\) that corresponds to your answer in part (a).
Evaluate \(\lim_{x \to 2^+} f(x)\), describing what happens to the curve as \(x\) approaches \(2\) from the positive side.
All 16 limits and rates of change questions with mark schemes →
FAQ
How many IB Maths AA SL limits and rates of change questions are there?
There are 16 exam-style limits and rates of change questions in the AA SL question bank (Paper 1: 9 · Paper 2: 7), graded 4 easy, 5 medium, 4 hard, 3 very hard. Every question has a full IB-style mark scheme (M, A and R marks).
Is limits and rates of change on Paper 1 or Paper 2?
Both. In the bank, Paper 1: 9 · Paper 2: 7. Paper 1 is non-calculator, so practise the exact-value algebra as well as the GDC methods.
Where can I get the mark schemes?
Open the AA SL Unit 5 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
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