L'Hôpital's Rule Calculator

The L'Hôpital's Rule Calculator resolves a limit of the form f(x)/g(x) as x approaches a point a, whenever direct substitution gives an indeterminate 0/0 or ∞/∞ form. It repeatedly differentiates the numerator and denominator until the form is no longer indeterminate, then evaluates the resulting ratio — showing the form found at each pass and the full chain of derivatives down to the final answer.

It's built for calculus students checking a limit by hand, and for anyone who needs to resolve an indeterminate limit without differentiating the same expression over and over on paper.

How to use it

1. Enter the numerator f(x) and denominator g(x) — standard math notation works: ^ for powers, * for multiplication, plus sqrt(), sin(), cos(), log(), and the other functions supported by mathjs's expression parser.
2. Choose the variable (x, y, z, or t) and enter the limit point a — a plain number, or Infinity / -Infinity for a limit at infinity.
3. Read Limit for the final answer, Indeterminate Form for what direct substitution produced (0/0, ∞/∞, or "not indeterminate" if the rule didn't need to run), and Rule Applications for how many times it differentiated.
4. Step-by-Step Derivation shows f(a) and g(a) at every pass, each differentiation, and the final evaluation.

Results update automatically as you type. For example, sin(x)/x as x → 0 is a 0/0 form; one application gives cos(x)/1, which evaluates to 1 — so the limit is 1. ln(x)/x as x → Infinity is an ∞/∞ form; one application gives (1/x)/1, which evaluates to 0 at infinity — so the limit is 0.

What is L'Hôpital's rule?

If f(a)/g(a) is 0/0 or ∞/∞ (an indeterminate form — direct substitution gives no real answer), then, provided the derivatives exist and g′ doesn't vanish near a:

lim(x→a) f(x)/g(x) = lim(x→a) f′(x)/g′(x)

If the new ratio is still indeterminate, the rule can be applied again to f″/g″, and so on — this calculator applies it automatically, up to 6 times, and stops as soon as substitution gives a genuine number (or diverges to ±∞).

Why check the form before differentiating?

L'Hôpital's rule only applies to a genuine 0/0 or ∞/∞ form — applying it to a limit that already evaluates directly gives a wrong answer, not just an unnecessary one. This calculator evaluates f(a) and g(a) separately (never as one ratio) before touching a derivative, so it only invokes the rule when the form actually justifies it, and reports "not indeterminate" plus the direct answer otherwise.

What functions can I enter?

Anything mathjs's expression evaluator understands: polynomials (x^2 - 1), trigonometric functions (sin(x)), roots, exponentials, and logarithms (sqrt(x), exp(x), log(x)), and combinations of these. Angles in trig functions are in radians by default — write pi for π.

To find a partial derivative of a multivariable function step by step, see the Partial Derivative Calculator.

Privacy

This calculator runs entirely in your browser. Your functions and limit point are never uploaded, logged, or stored — the computation happens locally on your device.