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 $f'(x)$ .

Substitute $x+h$ into $f(x) = x^2$ : $\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h}$ . (M1)A1 The $x^2$ terms cancel out: $\frac{2xh + h^2}{h}$ . Factor out an $h$ from the numerator: $\frac{h(2x + h)}{h} = 2x + h$ . A1 Take the limit as $h$ approaches $0$ : $\lim_{h \to 0} (2x + h) = 2x + 0 = 2x$ . (M1) Thus, $f^{\prime}(x) = 2x$ . A1

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IB Maths AA SL · Unit 5: Calculus

IB Maths AA SL Limits and Rates of Change Questions

Updated October 2026

Exam-style IB Maths AA SL limits and rates of change questions with worked solutions. Start with the 2 fully worked examples below — each is solved step by step, showing the working that earns the marks — then try the practice questions and check your working against the mark scheme.

41 questions
Paper 1: 31
Paper 2: 10
11 easy
15 medium
10 hard
5 very hard
2 worked examples

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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 (26)
Evaluating limits (10)
Average rates of change (5)

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 $lim_{x \to 3} f (x)$ .

Graph of y = f(x): a straight line from x = -1 to x = 5 with a hole at (3, 4) and a solid point at (3, 2)

Solution

6 steps · 1 to fill in

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: Limits at infinity and horizontal asymptotes · hard

Evaluate $lim_{x \to \infty} \frac{3 x + 1}{x - 2}$ and briefly explain the geometric significance of this limit regarding the graph of the rational function.

Solution

6 steps

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. One of them has its full worked solution here, open to everyone.

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 $f (x) = - x^{2} + 4 x + 5$ and its derivative is $f^{'} (x) = - 2 x + 4$ :

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.

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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 $f^{'} (x)$ .

Full worked solution Free, no sign-in

Substitute $x + h$ into $f (x) = x^{2}$ :
$\frac{(x + h)^{2} - x^{2}}{h} = \frac{x^{2} + 2 x h + h^{2} - x^{2}}{h}$ . (M1)A1
The $x^{2}$ terms cancel out: $\frac{2 x h + h^{2}}{h}$ .
Factor out an $h$ from the numerator: $\frac{h (2 x + h)}{h} = 2 x + h$ . A1
Take the limit as $h$ approaches $0$ :
$lim_{h \to 0} (2 x + h) = 2 x + 0 = 2 x$ . (M1)
Thus, $f^{'} (x) = 2 x$ . A1

Every question here has a full worked solution. See them all →

Question 3 · hard · 5 marks · Paper 1

Consider the rational function $f (x) = \frac{3 x + 1}{x - 2}$ , 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.

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FAQ

How many IB Maths AA SL limits and rates of change questions are there?

There are 41 exam-style limits and rates of change questions in the AA SL question bank (Paper 1: 31 · Paper 2: 10), graded 11 easy, 15 medium, 10 hard, 5 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: 31 · Paper 2: 10. 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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IB Maths AA SL Limits and Rates of Change Questions

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What's examined in AA SL limits and rates of change

Limits and Rates of Change worked examples

Worked example 1: Reading graphical limits and function values · easy

Solution

Worked example 2: Limits at infinity and horizontal asymptotes · hard

Solution

Try these IB Maths AA SL limits and rates of change questions

Question 1 · easy · 4 marks · Paper 1

Question 2 · medium · 5 marks · Paper 1

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Question 3 · hard · 5 marks · Paper 1

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