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Limit Calculator
Use this limit calculator to find the limit of a function as x approaches a point — or as x heads to infinity. Type in f(x), pick the point and the side, and the calculator evaluates the function at points closer and closer to that point, showing you the table of approaching values so you can see the limit emerge, or see honestly when it does not exist.
Type inf or -inf for infinity.
Allowed: numbers, x, + − * / ^ ( ) . and sin cos tan ln log sqrt abs exp. Write 2*x, not 2x. Log is base 10; ln is the natural log.
How it works
A limit asks where a function is heading, not what it equals at a point. Sometimes those are the same thing — for a smooth, unbroken function, the limit as x approaches a is simply f(a). But the interesting cases are the ones where f(a) is undefined, or where the function jumps: the limit still exists if the values approaching from both sides agree, even when the function itself has a hole or a jump at that exact point.
The function f(x) is what you type in. Use x as the variable and write multiplication out in full: 2*x, not 2x, and put parentheses around fractions, as in (x^2-1)/(x-1). Powers use ^, so x squared is x^2. The functions sin, cos, tan, ln, log, sqrt, abs, and exp are all available. For safety, the calculator only accepts those characters and function names — anything else is rejected with a message, and nothing you type is ever run as raw code.
The point a is where x is heading. Enter an ordinary number such as 1, 0, or 2.5. You can also type inf, +inf, or infinity for positive infinity, or -inf or -infinity for negative infinity, and the calculator will test the function at large values of x instead of near a single point.
The side chooses which direction x comes from. "Both sides" is the usual two-sided limit: it exists only when the left-hand limit and the right-hand limit agree. "From the left only" tests x values just below a, and "From the right only" tests x values just above a. One-sided limits are the right tool when a function behaves differently on each side of a point, or when it is only defined on one side.
The calculator evaluates f at a ± 0.1, a ± 0.01, a ± 0.001, and a ± 0.0001, and shows every value in the table. If the last two values on a side agree to within a small tolerance, that side has settled. If the values grow past about a million in size, the side diverges to infinity. If the two sides settle on different numbers, or head to opposite infinities, the two-sided limit does not exist — and the calculator says that plainly instead of inventing an answer. This is a numerical method, so it is an excellent check for homework and a good way to build intuition, but a formal proof still needs algebra: factor, cancel, and simplify by hand to confirm what the numbers suggest.
Worked example
Find the limit of f(x) = (x² − 1) ÷ (x − 1) as x approaches 1. Directly plugging in x = 1 gives 0 ÷ 0, which is undefined — so the limit has to be found by approaching 1 instead.
Step 1: Try x = 0.9, just left of 1. f(0.9) = (0.81 − 1) ÷ (0.9 − 1) = (−0.19) ÷ (−0.1) = 1.9.
Step 2: Try x = 0.99. f(0.99) = (0.9801 − 1) ÷ (−0.01) = 1.99.
Step 3: Try x = 0.999 and x = 0.9999. The values are 1.999 and 1.9999 — clearly heading for 2.
Step 4: Try the right side. f(1.1) = 2.1, f(1.01) = 2.01, f(1.001) = 2.001. Both sides agree.
Try it yourself: enter (x^2-1)/(x-1), set a to 1, choose "Both sides", and press Calculate. The tool gives exactly the same answer — the limit is 2 — and the table shows the values 1.9, 1.99, 1.999, and 1.9999 on the left and 2.1, 2.01, 2.001, and 2.0001 on the right. Algebra agrees: x² − 1 factors to (x − 1)(x + 1), the (x − 1) cancels, and you are left with x + 1, which is 2 at x = 1.
FAQ
What is a limit in calculus?
A limit is the value a function gets closer and closer to as x gets closer and closer to a point. It describes where the function is heading, even if the function is undefined exactly at that point. For example, (x squared minus 1) divided by (x minus 1) approaches 2 as x approaches 1, even though it is undefined at x = 1.
How does this limit calculator work?
It evaluates your function at points very close to the target from the left and from the right — at distances of 0.1, 0.01, 0.001, and 0.0001 — and watches what the values settle toward. If both sides settle on the same number, that is the limit. If they settle on different numbers, or grow without bound, the calculator says so.
When does a limit not exist?
A two-sided limit does not exist when the left-hand and right-hand limits disagree. For 1 divided by x at x = 0, the left side heads toward negative infinity and the right side toward positive infinity, so there is no single limit. A limit also fails to exist in the usual sense when the function grows without bound on both sides.
What functions can I type in?
You can type expressions in x using numbers, plus, minus, times, divide, powers with ^, and parentheses, plus the functions sin, cos, tan, ln, log, sqrt, abs, and exp. Write multiplication out in full — use 2*x, not 2x — and use (x^2-1)/(x-1) with parentheses around a fraction. Log means base 10 and ln means the natural log.
Can this calculator find limits at infinity?
Yes. Choose +infinity or -infinity as the point, and the calculator evaluates the function at x = 10, 100, 1000, and 10000 (or the negative of those) to see what it settles toward. For 1 divided by x at +infinity, the values 0.1, 0.01, 0.001, and 0.0001 settle toward 0, so the limit is 0.