Education
L'Hopital Calculator
Enter polynomial coefficients for the numerator and denominator, and the value x = a. The calculator evaluates the original ratio; if it is 0/0, it differentiates both polynomials and evaluates the new ratio.
Polynomial coefficients
Enter coefficients in descending order. For example, x − 1 is 1, -1; x² − 4 is 1, 0, -4.
e.g. 1, -1
e.g. 1, -1
e.g. 1
Scope
This calculator applies L'Hôpital's rule once to a polynomial-over-polynomial limit at a finite x = a. If direct substitution gives 0/0, the calculator differentiates both sides and evaluates the new ratio.
Out of scope: ∞/∞ forms, exponentials and logarithms, products requiring rewriting, and multiple L'Hôpital iterations. For ∞/∞ on rationals, use the limits-at-infinity calculator.
Limit (after L'Hôpital)
1
Indeterminate 0/0 at x = 1; differentiated and re-evaluated to 1.
L'Hôpital's rule applies only when direct substitution gives an indeterminate form like 0/0 or ∞/∞. When it does, you replace the limit of f/g with the limit of f'/g' (the derivative ratio).
Examples
lim (x − 1) / (x − 1) at x = 1
0/0; after rule: 1/1 = 1
lim (x² − 1) / (x − 1) at x = 1
0/0; after rule: 2x/1 at 1 = 2
lim (x² − 4) / (x² − x − 2) at x = 2
0/0; after rule: 4/3
lim (3x − 6) / (x² − 4) at x = 2
0/0; after rule: 3/4
How it works
L'Hopital's rule turns a hard 0/0 limit into a manageable derivative ratio. The polynomial derivative uses the power rule, applied term-by-term to the coefficient list.
Rule · lim f/g at a = lim f'/g' at a (when 0/0)
Power rule · d/dx (xⁿ) = n · xⁿ⁻¹
Related calculators
- Limit calculator for direct substitution and removable-discontinuity cancellation.
- Limits at infinity calculator for x → ±∞ on rational functions.
- Derivative calculator for polynomial differentiation in general.
- Integral calculator for the antiderivative side of calculus.
- All education calculators.
Frequently asked questions
If lim f(x)/g(x) gives an indeterminate form 0/0 or infinity/infinity at x = a, then under mild conditions the limit equals lim f'(x)/g'(x). The rule replaces a hard limit with an easier one.
Only on indeterminate forms 0/0 or infinity/infinity. If direct substitution gives a defined value (like 3/2), L'Hopital does not apply; the limit is simply that value.
Apply the rule again. The calculator applies the rule once. If the result is still 0/0, it flags the case and asks you to apply the rule again by hand or with a CAS.
Mathematically yes, but this calculator handles polynomial-over-polynomial inputs only. For trig, exp, and log limits, use a CAS or apply the rule by hand using the standard derivative rules.
Coefficient input is unambiguous and never breaks. A general symbolic parser is fragile and easy to get wrong. List the polynomial coefficients in descending order and the calculator handles the rest.
The limit calculator covers polynomial and rational expressions with direct substitution and one round of (x − a) cancellation. The L'Hopital calculator is specifically for 0/0 indeterminate forms where you want to see the derivative ratio step.
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