L’Hôpital’s Rule Calculator
Use this free L’Hôpital’s Rule Calculator to evaluate limits involving indeterminate forms like 0/0 or ∞/∞. Enter the numerator and denominator functions, specify the limit point, and get a step-by-step solution showing each application of L’Hôpital’s rule with an interactive graph.
Evaluate limits using L’Hôpital’s rule for indeterminate forms (0/0 or ∞/∞).
Result
Step-by-Step Solution
Graph
What is L’Hôpital’s Rule?
L’Hôpital’s Rule provides a method to evaluate limits that result in indeterminate forms. When direct substitution gives you \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), this rule lets you take derivatives of the numerator and denominator separately to find the limit.
Named after French mathematician Guillaume de l’Hôpital (though actually discovered by Johann Bernoulli), this technique transforms impossible-looking limits into solvable problems.
The Rule
If \( \lim \frac{f(x)}{g(x)} \) gives \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), then:
$$\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}$$
provided the limit on the right exists or is \( \pm\infty \).
Indeterminate Forms
Direct Forms: \( \frac{0}{0} \) and \( \frac{\infty}{\infty} \)
Both numerator and denominator approach the same type (both zero or both infinity). Apply L’Hôpital’s Rule directly.
Other Indeterminate Forms
Forms like \( 0 \cdot \infty \), \( \infty – \infty \), \( 0^0 \), \( 1^\infty \), and \( \infty^0 \) require algebraic manipulation first to convert them into \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) form.
Classic Examples
Example 1: \( \frac{\sin(x)}{x} \) as \( x \to 0 \)
Direct substitution gives \( \frac{0}{0} \). Apply the rule:
$$\lim_{x \to 0} \frac{\sin(x)}{x} = \lim_{x \to 0} \frac{\cos(x)}{1} = \frac{\cos(0)}{1} = 1$$
Example 2: \( \frac{e^x – 1}{x} \) as \( x \to 0 \)
Direct substitution gives \( \frac{0}{0} \). Apply the rule:
$$\lim_{x \to 0} \frac{e^x – 1}{x} = \lim_{x \to 0} \frac{e^x}{1} = \frac{e^0}{1} = 1$$
Example 3: \( x \cdot \ln(x) \) as \( x \to 0^+ \)
This is \( 0 \cdot (-\infty) \) form. Rewrite as \( \frac{\ln(x)}{1/x} \), which gives \( \frac{-\infty}{\infty} \). Apply the rule:
$$\lim_{x \to 0^+} \frac{\ln(x)}{1/x} = \lim_{x \to 0^+} \frac{1/x}{-1/x^2} = \lim_{x \to 0^+} (-x) = 0$$
When to Use L’Hôpital’s Rule
Use It When
- Direct substitution gives \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \)
- Both functions are differentiable near the point
- The derivative limit exists
Don’t Use It When
- The limit isn’t indeterminate (the rule gives wrong answers)
- Algebraic simplification works faster
- The derivative makes things more complex
- You can factor or rationalize instead
Common Mistakes
- Applying to non-indeterminate forms – The rule only works for \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \)
- Using the quotient rule – Take derivatives separately, not as a quotient
- Infinite loops – Sometimes applying the rule repeatedly doesn’t help. Try algebraic manipulation instead.
- Forgetting to check conditions – Always verify the form is indeterminate before applying
Converting Other Indeterminate Forms
| Form | Convert To | Method |
|---|---|---|
| \( 0 \cdot \infty \) | \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) | Rewrite as \( \frac{f}{1/g} \) |
| \( \infty – \infty \) | \( \frac{0}{0} \) | Find common denominator |
| \( 0^0, 1^\infty, \infty^0 \) | \( \frac{0}{0} \) | Take ln, then use \( e^{\ln(…)} \) |
What is L’Hôpital’s Rule used for?
L’Hôpital’s Rule is used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. When direct substitution doesn’t work because both numerator and denominator approach zero or infinity, this rule lets you take derivatives of each separately to find the limit.
When can you apply L’Hôpital’s Rule?
You can apply L’Hôpital’s Rule when: (1) direct substitution gives 0/0 or ∞/∞, (2) both the numerator and denominator are differentiable near the point, and (3) the limit of the ratio of derivatives exists. Never apply it to limits that aren’t indeterminate forms.
What are the 7 indeterminate forms?
The seven indeterminate forms are: 0/0, ∞/∞, 0·∞, ∞−∞, 0⁰, 1^∞, and ∞⁰. L’Hôpital’s Rule directly applies to 0/0 and ∞/∞. The other forms must first be converted to one of these two forms through algebraic manipulation before applying the rule.
Can you use L’Hôpital’s Rule multiple times?
Yes, you can apply L’Hôpital’s Rule repeatedly as long as each application still results in an indeterminate form (0/0 or ∞/∞). Check the form after each application. Sometimes repeated application leads to a solvable limit, but occasionally you’ll get stuck in a loop and need a different approach.
Why do you differentiate separately and not use the quotient rule?
L’Hôpital’s Rule specifically requires differentiating the numerator and denominator independently, not as a single quotient. Using the quotient rule gives incorrect results because the rule is based on a theorem about the ratio of derivatives, not the derivative of a ratio. This is a common mistake students make.
What if L’Hôpital’s Rule doesn’t seem to work?
If L’Hôpital’s Rule leads to increasingly complex expressions or cycles without converging, try a different approach: factor the expression, use trigonometric identities, rationalize, use Taylor series expansions, or algebraically simplify before applying the rule. The rule isn’t always the most efficient method.
How do you handle 0 times infinity form?
Convert 0·∞ to 0/0 or ∞/∞ by rewriting one factor as its reciprocal in the denominator. For example, x·ln(x) as x→0⁺ can be rewritten as ln(x)/(1/x), which gives -∞/∞. Then apply L’Hôpital’s Rule normally. Choose the form that makes differentiation easier.
Who actually discovered L’Hôpital’s Rule?
Johann Bernoulli discovered the rule, but it’s named after Guillaume de l’Hôpital. L’Hôpital was a French nobleman who paid Bernoulli for mathematical instruction and the rights to publish his discoveries. The rule first appeared in l’Hôpital’s 1696 textbook, the first calculus textbook ever published.
