L'Hôpital's Rule Calculator

What is L'Hôpital's Rule?

In calculus, L'Hôpital's rule (also written as **L'Hospital's rule**) is a method for evaluating limits of indeterminate forms. Named after the 17th-century French mathematician Guillaume de l'Hôpital, the rule states that under certain conditions, the limit of a quotient of two functions is equal to the limit of the quotient of their derivatives.

When you attempt to evaluate a limit $\lim_{x \to c} \frac{f(x)}{g(x)}$ directly and obtain an indeterminate form like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, L'Hôpital's rule allows you to differentiate the numerator and denominator separately to resolve the limit:

Prerequisites and Conditions

Before applying L'Hôpital's rule, you must verify that all of the following conditions are met:

1. Indeterminate Form: The limit must produce either $\frac{0}{0}$ or $\frac{\pm\infty}{\pm\infty}$. If evaluating the limit directly yields a determinate form (e.g. $\frac{5}{2}$ or $\frac{0}{3}$), the rule cannot be applied.
2. Differentiability: The functions $f(x)$ and $g(x)$ must be differentiable on an open interval around $c$ (except possibly at $c$ itself).
3. Non-Zero Denominator Derivative: The derivative of the denominator $g'(x)$ must not be zero on the interval around $c$ (except possibly at $c$).
4. Limit of Derivatives Exists: The limit $\lim_{x \to c} \frac{f'(x)}{g'(x)}$ must exist, or tend to $\infty$ or $-\infty$.

Common Indeterminate Forms

While the rule is formulated directly for $\frac{0}{0}$ and $\frac{\infty}{\infty}$, other indeterminate forms can often be rewritten using algebraic manipulation to fit these two standard forms:

• Indeterminate Product ($0 \times \infty$): Rewrite $f(x)g(x)$ as $\frac{f(x)}{1/g(x)}$ or $\frac{g(x)}{1/f(x)}$.
• Indeterminate Difference ($\infty - \infty$): Find a common denominator or factor out terms to express as a single fraction.
• Indeterminate Powers ($1^\infty$, $0^0$, $\infty^0$): Take the natural logarithm of the expression, evaluate the limit, and then exponentiate the result ($e^L$).

Step-by-Step Examples

Example 1: $\lim_{x \to 0} \frac{\sin(x)}{x}$

1. Substitute $x = 0$ directly: $\frac{\sin(0)}{0} = \frac{0}{0}$. This is indeterminate.
2. Differentiate the numerator: $\frac{d}{dx}[\sin(x)] = \cos(x)$.
3. Differentiate the denominator: $\frac{d}{dx}[x] = 1$.
4. Evaluate the new limit: $\lim_{x \to 0} \frac{\cos(x)}{1} = \frac{\cos(0)}{1} = \frac{1}{1} = 1$.

Example 2: $\lim_{x \to \infty} \frac{x^2}{e^x}$

1. Substitute $x \to \infty$ directly: $\frac{\infty}{\infty}$. This is indeterminate.
2. First application of L'Hôpital's: $\lim_{x \to \infty} \frac{2x}{e^x}$. This still yields $\frac{\infty}{\infty}$.
3. Second application of L'Hôpital's: $\lim_{x \to \infty} \frac{2}{e^x}$.
4. Evaluate: $\frac{2}{\infty} = 0$.

For related calculus tools, try the Derivative Calculator, Integral Calculator, or Linear Equation Solver.