Limit Calculator

How to Use the Limit Calculator

Enter your function

Type f(x) using ^ for powers and names like sin, cos, ln, exp — for example sin(x)/x or (x^2-1)/(x-1).

Set what the variable approaches

Type the value the variable heads toward — a number, pi, e, or inf / -inf for limits at infinity.

Choose a direction

Leave it on "Both sides" for the ordinary limit, or pick Left or Right for a one-sided limit — useful where the function jumps or blows up.

Read the result

The calculator returns the limit, tells you when it is infinite, and reports when a two-sided limit does not exist because the left and right sides disagree.

Limits: The Idea Calculus Is Built On

Getting Arbitrarily Close

A limit asks a deceptively simple question: as the input gets closer and closer to some value, what does the output get closer and closer to? It is not the same as just plugging the value in. Many of the most important limits occur exactly where direct substitution fails — where you would get zero divided by zero, or a value that is undefined. The classic example is sin(x) divided by x as x approaches zero: substitute zero and you get the meaningless 0/0, yet the function settles unmistakably toward 1 as you approach from either side. That number, 1, is the limit, and capturing this idea of "the value it is heading for" rigorously is the achievement that made calculus possible. Derivatives are defined as limits, integrals are defined as limits, and the entire edifice of continuous mathematics rests on this one concept. This calculator evaluates limits the way you would explore them by hand — by probing values closer and closer to the target and watching where the function goes — while preferring exact substitution whenever the function is simply continuous at the point.

There is real subtlety here that the calculator is built to respect. A two-sided limit exists only if the function approaches the same value from the left and from the right; where those disagree — as with 1/x at zero, which dives to negative infinity on one side and positive infinity on the other — the two-sided limit does not exist, and the tool says so and invites you to try a one-sided limit instead. Limits can also be infinite, describing a function that grows without bound, and they can be taken "at infinity" to describe a function's long-run behaviour, like the way (2x+1)/(x+3) settles toward 2 as x grows huge. The calculator handles all of these, and it does so carefully: numerical limit-finding is notoriously prone to floating-point error when you probe too close to the target, so the engine deliberately stops at a safe distance and snaps clean results to exact values, avoiding the cancellation mistakes that trip up naive implementations.

"A limit is the value a function heads toward — not necessarily the value it reaches. It is how mathematics captures 'arbitrarily close' with complete precision."

Why Limits Matter Beyond the Classroom

Limits are the bridge between the discrete and the continuous, and they appear wherever you ask what happens "in the limit". The derivative — the slope of a curve, the speed of a moving object — is defined as the limit of an average rate of change as the interval shrinks to zero. The definite integral — the area under a curve — is the limit of a sum of ever-thinner rectangles. Limits at infinity describe the steady state a system relaxes into, the horizontal asymptote a graph approaches, or the long-run rate of return on an investment. One-sided limits formalise jumps and discontinuities, which matter in everything from signal processing to economics, where a price or a policy can change abruptly. Even the famous number e arises as a limit, of (1 + 1/n) raised to the power n as n grows without bound. Whether you are learning calculus, checking that a tricky limit really does exist, or exploring how a function behaves near a trouble spot or far out toward infinity, this calculator gives you the answer instantly and privately, entirely in your browser.

10 Facts About Limits

01

A limit is the value a function approaches — not always the value it reaches.

02

The famous limit sin(x)/x → 1 as x → 0, despite being 0/0 on substitution.

03

The derivative is defined as a limit of an average rate of change.

04

A two-sided limit exists only if left and right agree.

05

Limits can be taken at infinity to describe long-run behaviour.

06

An infinite limit means the function grows without bound.

07

One-sided limits formalise jumps and discontinuities.

08

The number e is the limit of (1 + 1/n)ⁿ as n → ∞.

09

Cauchy and Weierstrass gave limits their rigorous ε–δ definition.

10

This tool probes safely to avoid floating-point cancellation errors.

Frequently Asked Questions

A limit is the single value a function gets arbitrarily close to as its input gets arbitrarily close to some target — even if the function is undefined exactly at that target. For example, sin(x)/x is undefined at x = 0, yet it heads toward 1 from both sides, so the limit is 1.
Type inf (or -inf) into the "Approaches" box. The calculator then examines the function's behaviour as the variable grows without bound — for instance, (2x+1)/(x+3) approaches 2 as x → ∞.
It is the value approached from only one direction — from the left (smaller values) or the right (larger values). Choose "Left" or "Right" in the "From" menu. One-sided limits are essential where a function jumps or where the two sides disagree, such as 1/x at zero.
A two-sided limit exists only if the function approaches the same value from both directions. When the left and right limits differ — as with 1/x at zero, which goes to −∞ on one side and +∞ on the other — the two-sided limit does not exist. You can still ask for each one-sided limit separately.
Yes. If the function grows without bound as the variable approaches the point — like 1/x² at zero — the limit is +∞ (or −∞). The calculator reports this and notes that the function is growing without bound.
The calculator evaluates limits numerically, but it prefers exact substitution where the function is continuous and snaps clean results to exact values like ½ or 1. It deliberately stops probing at a safe distance to avoid floating-point cancellation, so common textbook limits come out exactly.
Near zero, sin(x) is almost exactly equal to x, so their ratio is almost exactly 1 — and it gets closer to 1 the nearer x is to zero. Substituting zero gives the undefined 0/0, but the limit, which describes what the ratio approaches, is 1. This is one of the foundational limits of calculus.
No. The limit is evaluated entirely in your browser by a small built-in engine — nothing is uploaded to a server or a third-party library. It works offline once the page has loaded.
Yes — change the "Variable" box to any letter, such as t or n. The point it approaches and the function are interpreted in terms of that variable.
Completely free, with no account, sign-up, or limit on use. It runs entirely in your browser and collects no data. Evaluate as many limits as you like.

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