Introduction to Limits
A limit describes what a function approaches near a point.
About this topic
A limit describes the value a function approaches as the input gets close to some number, even if the function is not defined there. We write lim x→a f(x) = L. Estimate by tables of values or graphs from both sides. The limit exists only if the left and right limits agree.
The key idea is that the limit cares about what happens near a, not at a. The function f(x) = (x² − 1)/(x − 1) is undefined at x = 1, yet for x = 0.9, 0.99, 0.999 the outputs 1.9, 1.99, 1.999 approach 2, and since the function simplifies to x + 1 the limit is 2.
Limits fail to exist when the sides disagree, as with a jump, when the function grows without bound, or when it oscillates endlessly like sin(1/x) near zero. Limits are the foundation of calculus, defining derivatives, integrals and continuity, and they answer questions like what speed a car has at an exact instant.
Tips: always check from both sides, do not confuse the limit with the function value, and remember that a limit can be a perfectly good number even where the function has a hole.
Key ideas
- lim f(x) as x→a asks about nearby values
- The value at a point can differ from the limit
- Check from both sides
Quick check
Use cases: where Introduction to Limits helps
- Homework: review the key idea behind Introduction to Limits before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Introduction to Limits without notes.
- Visual learning: play with the interactive lab to see how Introduction to Limits changes when you adjust its inputs.
- Teaching a friend: explain Introduction to Limits in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Introduction to Limits?
A limit describes what a function approaches near a point.
What will I learn in Introduction to Limits?
lim f(x) as x→a asks about nearby values. The value at a point can differ from the limit. Check from both sides.
Is Introduction to Limits part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Introduction to Limits lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Introduction to Limits?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.