Limits
Lesson notes
Overview
A limit describes what a function approaches, even where it is not defined. This topic is rated advanced and is number 64 of 245 in Mathematics. Deeper models, formal results and harder reasoning.
What you are seeing
The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.
In depth
A limit describes the value a function approaches as its input gets close to a point, even if the function is not defined there. For example, sin x / x is undefined at x = 0 but tends to 1 as x shrinks towards zero. Limits make precise the ideas of instant, infinite and infinitely small that calculus relies on, and both the derivative and the integral are defined as limits. A limit can also fail to exist if the function approaches different values from the left and right.
Reading the relationship
The expression "lim x→0 sin x / x = 1" is a compact summary of the idea. It uses the symbols l, i, m, x; name what each one stands for before you start. Notice that it contains a ratio, so increasing the denominator makes the result smaller, and a trigonometric function, so the result repeats as the angle goes round. Then use the sliders to see how changing one quantity changes the others.
A guided exploration
1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.
Try this
Try to explain why the pattern holds, not just what it is. Look for limiting cases and what happens at the edges.
Common pitfalls
A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.
How it connects
It builds on “Derivative of eˣ”, the topic before it in Mathematics. Once you are comfortable, continue with “Continuity and Discontinuity”.
Why it matters
Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.
Where you might use it
- Revising “Limits” before a class, exam or certification
- Checking working on homework and exam questions by watching the graph respond
- Modelling growth, change and uncertainty in finance, data science and statistics
- Building the reasoning behind the algorithms and engineering tools people use every day
Frequently asked questions
What is Limits?
A limit describes what a function approaches, even where it is not defined. It is an advanced topic in Mathematics.
How do I use the Limits simulation?
The panel at the top shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. Move the sliders one at a time, then press Space to pause the animation and look closer.
What does the formula mean?
The expression lim x→0 sin x / x = 1 packs the idea into symbols. Name each symbol first, then change one quantity with the sliders and see which others respond.
Who is this topic for?
Deeper models, formal results and harder reasoning. If the quiz feels hard, reread the lesson notes and try the previous topic first.
What should I learn before this?
Start with “Derivative of eˣ”, the previous topic in Mathematics. Each topic in a subject builds on the ones before it.
How do I track my progress?
Press “Mark as learned” on this page. Progress is stored only in this browser and is never sent anywhere. The Browse page can show topics you have learned or still need to learn.