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Limit Calculator

Find the limit of a function of x as x approaches a number or infinity, with left and right limits, L’Hôpital’s rule and the steps.

Results

ItemValue

Steps

StepWorking
Type x as the variable, with ^ for powers and brackets for function inputs. The calculator tries direct substitution, then L’Hôpital’s rule for 0/0 and ∞/∞ forms, and checks every answer with values close to the point. Numeric results are rounded to 6 decimal places, so a limit like e is shown as 2.718282.

Quick Answer

A limit is the value a function gets close to as x gets close to a point. The limit of sin(x)/x as x → 0 is 1, found with L’Hôpital’s rule: cos(x)/1 = 1. The limit of (x^2 − 1)/(x − 1) as x → 1 is 2, and the limit of (1 + 1/x)^x as x → ∞ is e ≈ 2.718282.

Last updated: 2 November 2026 ·How we calculate this: direct substitution, L’Hôpital’s rule and one-sided checks

Table of Contents

How This Limit Calculator Works

Type a function of x and the value x approaches, such as 0, 2, pi or inf. The calculator first tries direct substitution. If that gives 0/0 or ∞/∞, it applies L’Hôpital’s rule by taking the derivative of the top and bottom. It also checks values just to the left and right of the point, so it can spot limits that go to infinity, jump, or do not exist.

Quick explanation: a limit asks where f(x) is heading, not what f(x) equals at the point.

  • Two-sided limit: exists only if the left and right limits match.
  • Infinite limit: f(x) grows without bound, written ∞ or −∞.
  • Limit at infinity: where f(x) heads as x gets very large.

How to Use the Limit Calculator

  1. Type your function using x as the variable.
  2. Enter the value x approaches, or inf for infinity.
  3. Choose both sides, left or right.
  4. Press Calculate.
  5. Read the limit, the left and right limits, and the method used.

Core Limit Rules

  • Direct substitution: if f is continuous at a, the limit is f(a)
  • L’Hôpital’s rule: for 0/0 or ∞/∞, lim f/g = lim f′/g′
  • Sum and product: lim (f + g) = lim f + lim g, and lim (f·g) = lim f · lim g
  • Special limit: lim sin(x)/x = 1 as x → 0
  • Special limit: lim (1 + 1/x)x = e as x → ∞

Key Terms

  • Limit: the value f(x) approaches as x approaches a point.
  • One-sided limit: a limit from only the left (x → a⁻) or the right (x → a⁺).
  • Indeterminate form: an expression like 0/0 or ∞/∞ that needs more work.
  • L’Hôpital’s rule: a rule that finds 0/0 and ∞/∞ limits using derivatives.
  • Continuous: a function with no breaks, where the limit equals the function value.

Worked Example

Find the limit of (x2 − 4)/(x − 2) as x → 2.

  • Substituting gives 0/0, which is undefined.
  • By L’Hôpital’s rule, take derivatives: 2x / 1.
  • At x = 2 this is 4, so the limit is 4. Factoring (x − 2)(x + 2)/(x − 2) = x + 2 gives the same answer.

Real-Life Examples

  • Holes in graphs: (x2 − 9)/(x − 3) has a hole at x = 3, but its limit is 6.
  • Vertical asymptotes: 1/x2 goes to ∞ as x → 0.
  • Horizontal asymptotes: (3x2 + 1)/(x2 − 5) goes to 3 as x → ∞.
  • Derivatives: the derivative is defined as a limit of slopes. See our Derivative Calculator.

Limit vs Function Value

  • Function value: what f(a) equals at the point, if it exists.
  • Limit: where f(x) heads as x approaches a.
  • Continuity: when both exist and match, the function is continuous at a.

Common Limits

LimitValue
sin(x)/x as x → 01
(1 − cos x)/x as x → 00
(ex − 1)/x as x → 01
(1 + 1/x)x as x → ∞e ≈ 2.718282
ln(x)/x as x → ∞0
1/x as x → 0⁺∞
1/x as x → 0⁻−∞
x·sin(1/x) as x → 00

Tips for Using the Calculator

  • Infinity: type inf or -inf.
  • One-sided limits: choose left or right, useful for 1/x at 0.
  • Constants: you can type pi or e as the point, such as pi/2.
  • Brackets: write (x^2 − 1)/(x − 1), with brackets around the top and bottom.

Common Mistakes to Avoid

  • Stopping at 0/0. 0/0 is an indeterminate form, not an answer. Simplify or use L’Hôpital’s rule.
  • Using L’Hôpital’s rule on other forms. It only works for 0/0 and ∞/∞.
  • Using the quotient rule. L’Hôpital’s rule differentiates the top and bottom separately.
  • Ignoring one side. A two-sided limit needs the left and right limits to match.
  • Confusing the limit with the value. A function can have a limit at a point where it is undefined.

Frequently Asked Questions

The value a function approaches as x approaches a certain number or infinity.

Try substituting the value. If you get 0/0, simplify, factor or use L’Hôpital’s rule.

The function does not settle on one value, for example when the left and right limits differ or it oscillates.

For 0/0 or ∞/∞ forms, the limit of f/g equals the limit of f′/g′, if that exists.

1.

Yes. If f(x) grows without bound, we write the limit as ∞ or −∞.

Type inf for positive infinity or -inf for negative infinity.

A limit as x approaches from only one side. Choose left or right in the calculator.

From the right it goes to ∞ and from the left to −∞, so the two sides differ.

Direct substitution and L’Hôpital answers are exact up to rounding. Other answers come from values very close to the point and are rounded to 6 decimal places.

Yes. Trig functions use radians.

As x → ∞, it is e, about 2.718282.

References & Sources

SourceUsed ForLink
OpenStax, Calculus Volume 1, Chapter 2: Limits Free textbook on limits, one-sided limits and limits at infinity. https://openstax.org/books/calculus-volume-1/pages/2-introduction
OpenStax, Calculus Volume 1, 4.8 L’Hôpital’s Rule Using derivatives to find 0/0 and ∞/∞ limits. https://openstax.org/books/calculus-volume-1/pages/4-8-lhopitals-rule