L'Hôpital's
Rule
Master the technique for evaluating limits involving indeterminate forms by differentiating the numerator and denominator.
What Is an Indeterminate Form?
When you substitute a limit value into a function and get an expression like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), the result is called an indeterminate form. These expressions don't tell you what the limit actually is: the true value could be any real number, \(\infty\), or may not exist at all.
Why Direct Substitution Fails
Consider \(\displaystyle\lim_{x \to 0} \frac{\sin x}{x}\). Plugging in \(x = 0\) gives \(\frac{0}{0}\), which is meaningless. Yet the limit equals 1. Direct substitution cannot resolve the "race" between numerator and denominator as both shrink toward zero.
The Big Idea Behind L'Hôpital's Rule
When \(\frac{f(x)}{g(x)}\) gives \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), both functions are approaching the same "finish line." The limit depends on how fast each function gets there: their rates of change.
L'Hôpital's Rule says: compare the derivatives instead. The ratio of the derivatives captures exactly the information that direct substitution lost.
L'Hôpital's Rule: Formal Statement
provided the right-hand limit exists (or is \(\pm\infty\)).
Three Conditions You Must Check
L'Hôpital's Rule vs. Quotient Rule
| L'Hôpital's Rule | Quotient Rule | |
|---|---|---|
| Purpose | Evaluate a limit | Find a derivative |
| What you do | Differentiate top and bottom separately | Apply \(\frac{f'g - fg'}{g^2}\) |
| When to use | Only for \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) | Any time you need \(\frac{d}{dx}\left[\frac{f}{g}\right]\) |
| Common trap | Applying the quotient rule by accident | Confusing it with L'Hôpital's |
How to Apply L'Hôpital's Rule
Example 1: \(\frac{0}{0}\) Form
Evaluate \(\displaystyle\lim_{x \to 0} \frac{\sin x}{x}\).
Example 2: \(\frac{\infty}{\infty}\) Form
Evaluate \(\displaystyle\lim_{x \to \infty} \frac{e^x}{x^2}\).
Example 3: A More Subtle Limit
Evaluate \(\displaystyle\lim_{x \to 0} \frac{e^x - 1 - x}{x^2}\).
Visualizing the Rate Comparison
Near \(x = 0\), the functions \(e^x - 1\) and \(x\) both approach 0, but at the same rate.
Other Indeterminate Forms
L'Hôpital's Rule only applies directly to \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\). For other indeterminate forms, rewrite first:
Example 4: Converting \(0 \cdot \infty\)
Evaluate \(\displaystyle\lim_{x \to 0^+} x \ln x\).
Common Mistakes to Avoid
Summary: When to Use What
| Limit Form | Strategy | Example |
|---|---|---|
| \(\frac{0}{0}\) | Apply L'Hôpital's directly | \(\frac{\sin x}{x}\) as \(x \to 0\) |
| \(\frac{\infty}{\infty}\) | Apply L'Hôpital's directly | \(\frac{e^x}{x^2}\) as \(x \to \infty\) |
| \(0 \cdot \infty\) | Rewrite as fraction, then apply | \(x \ln x\) as \(x \to 0^+\) |
| \(\infty - \infty\) | Combine fractions, then apply | \(\frac{1}{x} - \frac{1}{\sin x}\) |
| \(1^\infty, 0^0, \infty^0\) | Take \(\ln\), apply, exponentiate | \(x^x\) as \(x \to 0^+\) |
Question 1 of 5
What is \(\displaystyle\lim_{x \to 0} \frac{e^x - 1}{x}\)?
Question 2 of 5
Which indeterminate form does \(\displaystyle\lim_{x \to \infty} \frac{\ln x}{x}\) have?
Question 3 of 5
Evaluate \(\displaystyle\lim_{x \to 0} \frac{1 - \cos x}{x^2}\).
Question 4 of 5
A student applies L'Hôpital's Rule to \(\displaystyle\lim_{x \to 1} \frac{x^2 + 1}{x - 1}\). What error did they make?
Question 5 of 5
To evaluate \(\displaystyle\lim_{x \to 0^+} x \ln x\) using L'Hôpital's Rule, you should first rewrite it as: