L'Hôpital's Rule
Indeterminate Forms
Master the technique for evaluating limits that produce indeterminate forms like \(\frac{0}{0}\) and \(\frac{\infty}{\infty}\) by differentiating the numerator and denominator.
What Is an Indeterminate Form?
When you substitute the limiting value into a fraction and get \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), the limit is not determined by that substitution alone. The actual limit could be any real number, \(\infty\), \(-\infty\), or may not exist.
Why Direct Substitution Fails
Consider \(\displaystyle\lim_{x \to 0} \frac{\sin x}{x}\). Substituting \(x = 0\) gives \(\frac{0}{0}\), which tells us nothing. Yet the limit equals 1. The indeterminate form signals that the numerator and denominator vanish at the same rate.
The Intuition Behind the Rule
Near the point where both \(f(x)\) and \(g(x)\) equal zero, each function behaves approximately like its tangent line. So the ratio of the functions approximates the ratio of their slopes:
This is the core insight: replacing functions by their derivatives works because near the limit point, the derivative captures the local rate of vanishing or growth.
L'Hôpital's Rule: Formal Statement
The rule also applies when \(x \to a^+\), \(x \to a^-\), \(x \to \infty\), or \(x \to -\infty\). The result \(L\) may be a finite number, \(+\infty\), or \(-\infty\).
Conditions You Must Verify
Indeterminate vs. Determinate Forms
| Form | Type | L'Hôpital? |
|---|---|---|
| \(\frac{0}{0}\) | Indeterminate | ✓ Yes |
| \(\frac{\infty}{\infty}\) | Indeterminate | ✓ Yes |
| \(0 \cdot \infty\) | Indeterminate | Rewrite first |
| \(\infty - \infty\) | Indeterminate | Rewrite first |
| \(1^{\infty},\; 0^0,\; \infty^0\) | Indeterminate | Use \(\ln\) first |
| \(\frac{1}{0}\) | Determinate | ✕ No |
| \(\frac{0}{\infty}\) | Determinate (= 0) | ✕ No |
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 \(\displaystyle\lim_{x \to \infty} \frac{\ln x}{x}\).
Example 3: Repeated Application
Evaluate \(\displaystyle\lim_{x \to 0} \frac{1 - \cos x}{x^2}\).
Handling Other Indeterminate Forms
Some limits produce forms like \(0 \cdot \infty\) or \(\infty - \infty\). Rewrite them as fractions before applying L'Hôpital's Rule:
Example 4: \(0 \cdot \infty\) Form
Evaluate \(\displaystyle\lim_{x \to 0^+} x \ln x\).
Graphical Interpretation
When \(f(a) = g(a) = 0\), the ratio \(\frac{f(x)}{g(x)}\) near \(a\) is approximated by the ratio of their tangent-line slopes:
Common Mistakes to Avoid
Summary: When and How
| Scenario | Action | Form Needed |
|---|---|---|
| Direct sub gives \(\frac{0}{0}\) | Apply L'Hôpital | \(\frac{f'}{g'}\) |
| Direct sub gives \(\frac{\infty}{\infty}\) | Apply L'Hôpital | \(\frac{f'}{g'}\) |
| \(0 \cdot \infty\) | Rewrite as fraction | Then apply |
| \(\infty - \infty\) | Combine fractions | Then apply |
| \(1^\infty, 0^0, \infty^0\) | Take \(\ln\), rewrite | Then apply |
| Determinate form | Do NOT apply | Not needed |
Question 1 of 5
Evaluate \(\displaystyle\lim_{x \to 0} \frac{e^x - 1}{x}\).
Question 2 of 5
Evaluate \(\displaystyle\lim_{x \to \infty} \frac{x^2}{e^x}\).
Question 3 of 5
For which limit is L'Hôpital's Rule not applicable directly?
Question 4 of 5
Evaluate \(\displaystyle\lim_{x \to 0} \frac{\tan x - x}{x^3}\).
Question 5 of 5
Evaluate \(\displaystyle\lim_{x \to 0^+} x^x\).