L'Hôpital's Rule Calculator
Use L'Hôpital's rule to evaluate indeterminate limits.
Table of Contents
How to Use
- Enter the numerator and denominator as functions of the same variable (e.g., sin(x) and x).
- Set the variable name and the value it approaches.
- Run the calculation to check the direct ratio and, when needed, the derivative ratio.
- Compare both outputs to confirm if L'Hôpital's rule resolves the limit.
When to use L'Hôpital's rule
L'Hôpital's rule helps resolve indeterminate limits of type 0/0 or ∞/∞ by differentiating the numerator and denominator separately. If the new ratio has a finite limit, it matches the original limit.
- Confirm the limit has the form 0/0 or ∞/∞ at the approach point.
- Differentiate numerator and denominator individually.
- Evaluate the new ratio; repeat differentiation if needed.
- If the derivative denominator stays zero, the rule may not resolve the limit.
How this calculator approximates derivatives
- Evaluate f(x) and g(x) near the approach point to test for an indeterminate form.
- Use a small symmetric step to approximate f'(x) and g'(x).
- Compute the derivative ratio when the original ratio is indeterminate.
- Report both ratios so you can confirm whether L'Hôpital applies.
Numerical derivatives provide intuition but may differ from exact symbolic derivatives. Use algebraic checks for rigorous proofs.
Frequently Asked Questions
- What if the original limit is not 0/0 or ∞/∞?
- If the denominator is nonzero at the approach point, the calculator reports the direct ratio. L'Hôpital's rule is only needed for indeterminate forms.
- Why is the derivative ratio undefined?
- If the numerical derivative of the denominator is effectively zero, the derivative ratio cannot be computed. Try simplifying the functions or checking for higher-order applications of the rule.
- Does this replace symbolic differentiation?
- No. The calculator uses numeric derivatives for speed and intuition. For exact work, differentiate symbolically and then evaluate the resulting limit.