Calculadora da Regra de L’Hôpital (0/0)
- Created by
- Renato Passos, Eng. de Software
- Reviewed by
- Renato Passos, Eng. de Software
Last updated: Apr 18, 2026
Formula
lim f/g = lim f'/g' (quando f(a)=g(a)=0)
About this calculator
This numerical calculator approximates the limit of a ratio f(x)/g(x) as x approaches a value a, where both f(a) and g(a) are zero (0/0 indeterminate form). Instead of applying L'Hôpital's rule symbolically, it numerically computes the derivatives f'(a) and g'(a) using finite differences and returns the quotient f'(a)/g'(a). This is useful for quickly checking limits in calculus problems without manual differentiation.
L'Hôpital's rule states that, under certain conditions, the limit of f/g equals the limit of f'/g'. When both lateral limits exist and g'(a) ≠ 0, the approximate value is simply f'(a)/g'(a). The calculator uses a small step (h = 10⁻⁶) to estimate derivatives via central difference: f'(a) ≈ [f(a+h) - f(a-h)]/(2h). The result is a numerical approximation, not an exact limit, but generally accurate for well-behaved functions.
Use this tool in study or work situations where you need to compute limits of 0/0 indeterminate forms, such as in calculus exercises, function analysis, or mathematical modeling. It is especially useful when functions are complicated to differentiate symbolically or when you want a quick verification. Remember that L'Hôpital's rule only applies when f and g are differentiable near a and g'(a) ≠ 0.
Caution: the numerical approximation may fail if the derivative of g is zero at the point, resulting in division by zero or unstable values. Also, for functions with oscillatory behavior or discontinuities, the result may be inaccurate. Whenever possible, confirm the result with analytical methods or other tools. This calculator does not replace conceptual understanding of L'Hôpital's rule.
Frequently asked questions
What if the denominator g'(a) is zero?
If g'(a) = 0, the calculator will return an error or undefined value. In this case, L'Hôpital's rule does not apply directly; you may need to apply the rule again (if the derivatives are also zero) or use another method.
Does the calculator work for limits at infinity?
No, this calculator is designed for limits at a finite point a. For limits at infinity, a variable transformation or another approach would be needed.
How accurate is the approximation?
Accuracy depends on the function and the step size h used (10⁻⁶). For smooth functions, the error is of order h², about 10⁻¹². Functions with high curvature or singularities may have larger error.
Can I use it for indeterminate forms other than 0/0?
The calculator assumes both f(a) and g(a) are zero. For ∞/∞ forms, L'Hôpital's rule still applies, but the numerical approximation may not work well. A specific calculator for ∞/∞ is recommended.