Free IB Maths AA SL lesson: Informal limits
A ready-to-teach IB Maths AA SL lesson on Informal limits (Unit 5 · Calculus): editable slides and teacher notes, free; a printable worksheet in 3 levels with answers for school plans. About 55 minutes.
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What's inside
- Slides (PowerPoint, editable) · freeabout 12 slides: do now, objectives, key ideas, a worked example step by step, “your turn”, practice QR code, exit ticket
- Teacher notes (PDF) · freelesson plan with timings, common mistakes, homework links
- Worksheet (PDF) · school plans4 questions, 14 marks: Fluency, Reasoning, Exam-style. Included with school plans and the teacher trial.
- Answers (PDF) · school plansthe answer to every worksheet question, from its mark scheme. Included with school plans and the teacher trial.
Learning objectives
- Limits and differentiation from first principles
Key ideas on the slides
- The derivative is the limit of the gradient of a chord: f′(x) = lim_(h → 0)((f(x + h) − f(x))/h).
- Example: f(x) = x²: ((x + h)² − x²)/h = (2xh + h²)/h = 2x + h → 2x as h → 0.
- Average rate of change from x = a to x = b is (f(b) − f(a))/(b − a) (a chord); the instantaneous rate of change is f′(a) (a tangent).
- In a first-principles proof, write the limit notation on every line until h is removed, and do not let h = 0 before dividing.
By the end of the lesson, students should be able to answer a question like: Find the limit as x approaches 2 of (x^2 - 4)/(x - 2).
Key vocabulary
- limit: The value an expression approaches as the variable approaches a value.
- chord: A straight line joining two points on a curve.
- derivative: f′(x): the gradient function, the instantaneous rate of change.
Common mistakes this lesson tackles
- Substituting h = 0 before simplifying, getting 0/0.
- Dropping "lim_(h → 0)" from the working.
- Confusing average and instantaneous rates of change.
The worked example
Evaluate lim_(x → ∞) (3x + 1)/(x − 2) and briefly explain the geometric significance of this limit regarding the graph of the rational function.
- Strategy: divide every term by the highest power of x in the denominator (x¹).
- Divide: lim_(x → ∞) (3x/x + 1/x)/(x/x − 2/x).
- Simplify: lim_(x → ∞) (3 + 1/x)/(1 − 2/x).
- Evaluate: as x → ∞, 1/x → 0 and 2/x → 0.
- Substitute: (3 + 0)/(1 − 0) = 3.
- Explain: the limit is 3, which proves there is a horizontal asymptote at y = 3.
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 (3/1 = 3).
Calculator tips
- The d/dx key checks the value at a point; a table of (f(a + h) − f(a))/h for small h shows the limit.
Lesson plan
| 0–5 min | Do now: students write down what they remember about Venn diagrams & combined events and Inverse normal (finding cut-off values) |
| 5–8 min | Share the learning objective and success criteria |
| 8–20 min | Teach: the worked example, step by step; students copy the method |
| 20–25 min | Your turn: 1 mini-whiteboard question; check every board before moving on |
| 25–42 min | Practise: the worksheet (4 questions in 3 levels, with school plans and the teacher trial) or the topic's practice questions online; move up a level after two right in a row |
| 42–47 min | Plenary: 1-question exit ticket; collect it to plan the next lesson |
For students: practise this topic
The worksheet's QR code opens the IB Maths AA SL Unit 5: Limits and Rates of Change practice questions.
What schools get
- Set this lesson as homework, marked by AI against the mark scheme, with every result in your markbook (with a School plan).
- This lesson's worksheet in three levels, with its answers, ready to print.
- Support, Core and Stretch versions and A/B copies of the worksheet from the full question bank, in the worksheet builder.
- Create lesson: this lesson rebuilt for your class's next scheme-of-work lesson, from the full bank.
School licences from €490 a year. Try everything free for 30 days as a teacher: no card.
Questions teachers ask
Is this lesson free?
The slides and the teacher notes are free to download with a free IB Math Revision teacher account (sign in with Google, Microsoft or email); no card. The printable worksheet in three levels and its answers are included with school plans and the free 30-day teacher trial.
Can I edit the slides?
Yes. The slides are an ordinary PowerPoint file: change, add or delete anything. They also open in Google Slides and Keynote. The logo and web address sit on the slide master, so your edits do not move them.
What do students use for homework?
The practice slide's QR code and web address take students to the IB Maths AA SL Unit 5: Limits and Rates of Change practice questions on ibmathrevision.com, where the free questions are open and the full bank comes with a plan.