Limit Evaluator (Polynomial Ratio)

lim x→a of polynomial ratios (basic substitution + L'Hopital for 0/0).

Result

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How to use this calculator

Enter numerator + denominator coefficients.
Enter a (the value x approaches).

About this calculator

Limit of polynomial ratio: substitute directly if denominator non-zero. If 0/0 indeterminate: L'Hôpital's rule (differentiate numerator + denominator separately). Other indeterminate forms: ∞/∞, 0·∞, ∞−∞, 0⁰, ∞⁰, 1^∞ — also handled by L'Hôpital after rewriting. For polynomials at finite a, this calc covers basic + 1× L'Hôpital. Source: Wolfram MathWorld - Limit.

Frequently asked

L'Hôpital's rule?+

For 0/0 or ∞/∞: lim f/g = lim f'/g' (under conditions).

When NOT to use L'Hôpital?+

When form isn't 0/0 or ∞/∞. Don't apply blindly.

Limit at infinity?+

For rational functions: ratio of leading coefficients (if same degree); 0 (if denom higher); ±∞ (if numer higher).

One-sided limits?+

lim x→a⁺ (from above) vs. lim x→a⁻ (from below). Two-sided limit exists only if both equal.

Continuous functions?+

For continuous f: lim x→a f(x) = f(a). Direct substitution works.

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