Skip to main content
Calculator Answers

Limits at Infinity Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A limits at infinity calculator is a calculus tool that evaluates the limit of a mathematical function as the independent variable approaches positive or negative infinity. The calculator analyzes the behavior of rational functions, exponential equations, and radical expressions, applying calculus rules such as dividing by the highest power in the denominator or using L'Hopital's rule. It determines if the limit converges to a finite horizontal asymptote value or diverges to infinity. Calculus students use this utility to analyze asymptotic curve behavior.

Enter the leading coefficient and degree of the numerator and denominator, plus the direction. The calculator returns the limit using the standard degree-comparison rule.

See also: L'Hôpital Calculator and End Behavior Calculator.

Quick Answer

Calculate limits of functions as the variable approaches positive or negative infinity. Enter your mathematical function to find horizontal asymptotes and behavior.

Rational function leading terms

For a rational function P(x) / Q(x), only the leading terms of P and Q determine the limit at infinity. Enter the leading coefficient and degree for each.

Numerator: a · xⁿ + (lower terms)

Denominator: b · xᵐ + (lower terms)

Direction

For finite x → a limits, use the limit calculator. For 0/0 forms at a finite point, use the L'Hôpital calculator.

Limit at infinity

Limit

1.5

Numerator degree = denominator degree (2). The limit equals the ratio of leading coefficients: 3 / 2 = 1.5.

Numerator3·xⁿ with n = 2
Denominator2·xᵐ with m = 2
Directionx → +∞
Comparisonn = m → a/b

Only leading terms matter at infinity: every lower-order term shrinks relative to the leading term, so the comparison rule (n vs m) controls the answer.

Was this helpful?

Examples

lim (3x² − 2x) / (2x² + 1) as x → ∞

= 3/2 = 1.5

lim x³ / x² as x → ∞

= +∞

lim x² / x³ as x → ∞

= 0

lim −x³ / x² as x → −∞

= +∞ (signs combine)

How it works

For large |x|, only the leading term of each polynomial matters. So (a · xⁿ + …) / (b · xᵐ + …) behaves like (a/b) · x^(n − m) at infinity.

n < m · limit = 0

n = m · limit = a / b

n > m · limit = ±∞ (sign from a/b and direction)

Related calculators

Related Calculators

More tools from Education

L'Hôpital CalculatorApply L'Hôpital's rule to a polynomial-over-polynomial limit at x = a that gives 0/0 by direct substitution. Differentiates numerator and denominator, then evaluates the new ratio.End Behavior CalculatorDetermine the limits of a polynomial function as x approaches positive and negative infinity using the leading coefficient test.

Frequently asked questions

Compare the degrees of the numerator and denominator. If the numerator's degree is lower, the limit is 0. If they are equal, the limit is the ratio of leading coefficients. If the numerator's degree is higher, the limit is ±∞ depending on signs and direction.

As x grows, the leading term of any polynomial dominates the rest. So a · xⁿ / (b · xᵐ) behaves like (a/b) · x^(n−m) for large |x|. The sign of x^(n−m) determines whether the result is 0, finite, or infinite.

Same comparison, but x^(n−m) flips sign if (n − m) is odd. The calculator handles this by combining the sign of a/b with the direction and the parity of (n − m).

They do not change the answer at infinity (only the leading terms matter). For correctness, the calculator asks only for those.

For exponential, logarithmic, or transcendental functions, the leading-term rule does not apply. Use a CAS or apply L'Hôpital after rewriting. This page covers rational polynomials only.