What does x → ∞ mean in a limit?

It means x keeps growing past every number you could name, so ∞ is a direction to follow, not a value to plug in. The limit asks what the function settles toward on the way. As x grows, 1/x shrinks toward 0. By contrast, 12x divided by 4x equals 3 for every x except 0, so its limit at infinity is 3.

How do you solve a limit with a difference of cubes?

Factor the top with the pattern a³ − b³ = (a − b)(a² + ab + b²). In (x³ − 64) / (x − 4) as x → 4, that gives (x − 4)(x² + 4x + 16). Cancel the (x − 4), then substitute: 16 + 16 + 16 = 48. For this shape there is a quick check: three times the square of the number x approaches, 3 × 4² = 48.

When do students learn limits?

In many school systems, students meet limits at about 16 to 18, near the end of secondary school, and again in a first university calculus course. Lessons often follow the order of the levels here: plug in first, then untangle 0/0 forms, then take limits at infinity. These levels are set for ages 17 to 18.

Why is my answer wrong after cancelling in a limit?

Usually the cancelling is fine and the last step is missing: what remains must still be evaluated at the point. For (x² − 13x + 36) / (x − 9) as x → 9, the top is (x − 9)(x − 4). Cancelling leaves x − 4, so the limit is 9 − 4 = 5. Typing 4, the number left inside the bracket, is the classic slip.

Is this limit game free, and how could a calculus class use it?

Yes, all three levels are free for a class and open in any browser tab, with no accounts or installs. For calculus limits practice, write (x² − 9) / (x − 3), (x³ − 27) / (x − 3) and (x² − 5x + 6) / (x − 3) on the board. All three give 0/0 at x = 3, yet groups that factor them find limits of 6, 27 and 1. With the schools plan, charged per student, a teacher dashboard shows who is getting faster at limits.

Math practice · reimagined

Limits Game

This limit game sends x toward a number from 1 to 9 or off to infinity, and every answer, 0/0 traps included, is a whole number from 1 to 55. Spot where the function is heading before your rival, and your fighter strikes first.

Ages 4–99+ · no download · plays in your browser · 23 languages

Choose your level ▸

What is a limit in calculus?

A limit is the value a function closes in on as x gets closer and closer to a number. Writing lim x→3 means “as x approaches 3”, and the function need not even have a value at 3. Walk toward a door, each step halving the gap: you get as close as you like, and the door is your limit.

Every level of Limit

Example: the limit of (x² − 11x + 28) / (x − 7) as x → 7

Plug in 7 first: the top gives 49 − 77 + 28 = 0 and the bottom gives 7 − 7 = 0, so you get 0/0, which settles nothing.
Look for two numbers that multiply to 28 and add to 11: they are 7 and 4, so the top can be written as the product (x − 7)(x − 4).
Cancel the (x − 7) that sits on both the top and the bottom, which leaves only x − 4.
Now substitute 7 into x − 4 to get 3, so the limit is 3 and you type 3.

Learn it step by step

A limit asks where a function is HEADED as x creeps toward a value — the height it's aiming for.

When the function is well-behaved, you can just plug the value in. lim(x→2) of (x² + 1) = 2² + 1 = 5. Substitute and evaluate.

Type the value it approaches.

Tips and common mistakes

Follow one order every time: substitute first, and a number means you are done. A 0/0 means factor and cancel. For x → ∞, compare only the highest powers.
A common slip is cancelling the x’s in (x² − 25) / (x − 5) one by one. Only whole factors cancel, so first rewrite the top as (x − 5)(x + 5).
At home, a calculator makes a limit visible: put x = 3.9, 3.99 and 3.999 into (x² − 16) / (x − 4). The results, 7.9, 7.99 and 7.999, creep toward 8.

How to play

Each round shows one limit, such as lim x→4 (x² + 3x + 4), and you type the value it approaches, 32, on the keypad. Later levels bring 0/0 traps like (x² − 81) / (x − 9) as x → 9, and limits as x → ∞, but every answer is still a positive whole number. Answer first and Onion strikes, harder the faster you were.

— Quick answers —

Limit: the short FAQ

What does x → ∞ mean in a limit?: It means x keeps growing past every number you could name, so ∞ is a direction to follow, not a value to plug in. The limit asks what the function settles toward on the way. As x grows, 1/x shrinks toward 0. By contrast, 12x divided by 4x equals 3 for every x except 0, so its limit at infinity is 3.
How do you solve a limit with a difference of cubes?: Factor the top with the pattern a³ − b³ = (a − b)(a² + ab + b²). In (x³ − 64) / (x − 4) as x → 4, that gives (x − 4)(x² + 4x + 16). Cancel the (x − 4), then substitute: 16 + 16 + 16 = 48. For this shape there is a quick check: three times the square of the number x approaches, 3 × 4² = 48.
When do students learn limits?: In many school systems, students meet limits at about 16 to 18, near the end of secondary school, and again in a first university calculus course. Lessons often follow the order of the levels here: plug in first, then untangle 0/0 forms, then take limits at infinity. These levels are set for ages 17 to 18.
Why is my answer wrong after cancelling in a limit?: Usually the cancelling is fine and the last step is missing: what remains must still be evaluated at the point. For (x² − 13x + 36) / (x − 9) as x → 9, the top is (x − 9)(x − 4). Cancelling leaves x − 4, so the limit is 9 − 4 = 5. Typing 4, the number left inside the bracket, is the classic slip.
Is this limit game free, and how could a calculus class use it?: Yes, all three levels are free for a class and open in any browser tab, with no accounts or installs. For calculus limits practice, write (x² − 9) / (x − 3), (x³ − 27) / (x − 3) and (x² − 5x + 6) / (x − 3) on the board. All three give 0/0 at x = 3, yet groups that factor them find limits of 6, 27 and 1. With the schools plan, charged per student, a teacher dashboard shows who is getting faster at limits.

More math to master

Ready to throw down?

Play free ▸