Education
Limits at Infinity Calculator
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.
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
1.5
Numerator degree = denominator degree (2). The limit equals the ratio of leading coefficients: 3 / 2 = 1.5.
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.
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
- Limit calculator for finite x → a limits (substitution, removable discontinuities, numeric check).
- L'Hôpital calculator for 0/0 forms at a finite point.
- Scientific calculator for general arithmetic checks.
- All education calculators.
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.
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