# Limit Calculator - One-Sided Limits and Limits at Infinity

> Free limit calculator: lim x→a and x→±∞, left- and right-hand limits, 0/0, ∞/∞ and 1^∞ forms by series expansion; spots limits that do not exist.

- URL: https://qilingtools.com/en/math/limit
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Enter a function f(x) and the point a — a number such as 0, pi or 2/3, or inf / -inf for ±∞ — and choose a two-sided, left-hand or right-hand limit. The calculator returns the value, +∞ or −∞, or tells you that the limit does not exist because the one-sided limits differ or the function oscillates. It first uses series expansion: x is written as a + t (or 1/t at infinity) and every step of the expression is evaluated on Laurent series, so indeterminate forms such as sin x / x, (1 + 1/x)^x, (eˣ − 1)/x, (1 − cos x)/x² and √(x² + x) − x cancel exactly and the leading term (the asymptotic equivalent) is reported. For functions the series cannot handle, such as x·sin(1/x), xˣ or e^(1/x), it switches to a numerical approach with extrapolation and an error estimate. A table of values from each side and a graph near a help you see why.

## How to use

1. Enter f(x), for example sin(x)/x, (1+1/x)^x or x*sin(1/x)
2. Enter the point a (inf for +∞, -inf for −∞) and choose two-sided, left or right
3. Read the limit (shown exactly as a fraction, e, π and so on when recognised), the method, the table of values and the graph

## Features

- Limits as x → a, x → a⁻, x → a⁺ and x → ±∞
- Laurent-series leading-term analysis resolves 0/0, ∞/∞, 0·∞, ∞ − ∞ and 1^∞ forms
- Reports the leading term f(x) ≈ c·(x − a)ᵏ, i.e. the order of the zero or pole
- Computes left- and right-hand limits and flags a limit that does not exist when they differ
- Detects bounded and unbounded oscillation and functions defined on one side only (√x, ln x)
- Falls back to numerical sampling with Aitken extrapolation and an error estimate
- Recognises exact values: fractions, e, π, √2, ln 2, rational powers of e
- Table of values from each side, a graph near a, and a link to the grapher

## FAQ

### When does a limit not exist?

Three common cases: the left- and right-hand limits differ, as for 1/x or |x|/x at 0; the function oscillates without settling, as for sin(1/x) at 0; or the function is not defined near a. A limit of +∞ or −∞ also technically does not exist, but it is usually written lim = ∞ to describe the behaviour, and the calculator shows +∞ or −∞ directly.

### How does it handle 0/0 limits without L’Hôpital’s rule?

Instead of differentiating numerator and denominator repeatedly, it substitutes x = a + t and evaluates every operation on power series, so a common factor of t cancels inside the division. The lowest-order term c·tᵏ then decides the answer: k = 0 gives c, k > 0 gives 0 and k < 0 gives ±∞. This is the Taylor-series / equivalent-infinitesimal method, done automatically.

### Why are some results marked as a numerical estimate?

Functions with an essential singularity or a logarithmic term at a — x·sin(1/x), xˣ, e^(1/x), ln x / x — have no power series there. The calculator then samples over a hundred points approaching a on a logarithmic scale, decides between convergence, divergence and oscillation, and extrapolates with Aitken’s method. The result is approximate, an error estimate is shown, and common constants like 0, 1 and e are recognised.

### How do I enter a limit at infinity?

Type inf, +inf or ∞ for +∞ and -inf for −∞. For example (1+1/x)^x → e as x → +∞, (x^2+1)/(2x^2-3) → 1/2, and e^x → 0 as x → −∞. A limit at infinity has only one side, so the direction buttons are hidden.

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