Grade 12 · Mathematics
Limits
Why it matters
Approaching a value.
What is a limit in calculus?
Short answer
A limit describes the value a function approaches as its input approaches some number, whether or not the function is actually defined there. Written lim(x→a) f(x) = L, it means f(x) gets arbitrarily close to L as x gets close to a. Limits are the foundation of continuity, derivatives, and integrals.
Concept
What you will be able to do
- 01Define and explain limits
- 02Apply the key ideas of limits to solve problems
- 03Recognise common patterns and avoid typical mistakes
Concept
Concept
Approaching a value. In this lesson we build up limits step by step, starting from what you already know and connecting each idea to the next.
Understanding limits matters because it appears throughout this course and in later topics. Work through the examples slowly and try the quick-check questions before moving on.
Concept Lab
Explore the relationship live
Drag, zoom, and edit the functions without leaving the lesson.
Worked example
Follow the reasoning, not only the answer
Worked example 01
Worked example: apply limits to a typical problem.
- 1Write down what you know and what you need to find.
- 2Choose the right relationship or method.
- 3Substitute values carefully, keeping units.
- 4Solve and state the final answer clearly.
Mathematical conclusion
Final answer with correct units.
Common mistake
A common mistake is skipping units or the setup step — always show your reasoning.
Try it · retrieve before revealing
Check your understanding
Q1In one sentence, what is limits?
Check your answer against the concept explanation above.
Q2Give one situation where limits is used.
Compare with the worked example.
Alternative format
Listen to this lesson
Summary
Key ideas to carry forward
- ✓Limits builds directly on earlier ideas in this unit.
- ✓Practise a few questions to lock it in before the quiz.
What to practise next
Take the topic quiz, then try the short-answer set to practise writing full solutions.
Lesson formulas and key ideas
Formulas
Key idea
Key ideas
- Limits builds directly on earlier ideas in this unit.
- Practise a few questions to lock it in before the quiz.
Prove this skill
You’ve read it — now show it. Adaptive practice on this exact skill, with fresh problem types, until it is unaided and automatic. Reading a lesson never counts as mastery.
Content
Mark this lesson complete
Tracks what you have worked through — not mastery.
Mastery
Not yet demonstrated
Reading shows you have seen it. Prove you can do it — mastery is earned by answering questions unaided.
Related topics
Frequently asked questions
- Does a function have to be defined at a point to have a limit there?
- No. A limit describes the approach to a point; the function can be undefined exactly at that point and still have a limit.
- What is a one-sided limit?
- The value a function approaches from only the left (x→a⁻) or only the right (x→a⁺). A two-sided limit exists only when both agree.