L'Hospital's Rule for Quotients

5 min read

A Shortcut Through the Indeterminate

When both numerator and denominator approach zero (or both approach infinity), direct substitution fails. Before L’Hospital’s Rule, you might spend considerable effort factoring, rationalizing, or finding clever algebraic tricks.

L’Hospital’s Rule offers a systematic approach: differentiate the top and bottom separately, then take the limit again. It is elegant, powerful, and remarkably efficient when used correctly.

But beware: this rule only applies to $\frac{0}{0}$ and $\frac{\infty}{\infty}$ forms. Using it on other forms gives wrong answers. Always verify the form first!

Prerequisite Map

Quick Reference

Key Concepts

L’Hospital’s Rule

Suppose $f$ and $g$ are differentiable, $g'(x) \neq 0$ near $a$ (except possibly at $a$), and either:

$$\lim_{x \to a} f(x) = 0 \text{ and } \lim_{x \to a} g(x) = 0 \quad \text{(form } \tfrac{0}{0}\text{)}$$

or

$$\lim_{x \to a} f(x) = \pm\infty \text{ and } \lim_{x \to a} g(x) = \pm\infty \quad \text{(form } \tfrac{\infty}{\infty}\text{)}$$

Then:

$$\boxed{\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}}$$

provided the limit on the right exists (or equals $\pm\infty$).

Critical Warnings

Why It Works: The Geometric Intuition

Near the point where both $f(a) = 0$ and $g(a) = 0$, both functions look nearly linear (if we zoom in far enough). Their ratio is approximately:

$$\frac{f(x)}{g(x)} \approx \frac{f'(a)(x-a)}{g'(a)(x-a)} = \frac{f'(a)}{g'(a)}$$

The $(x-a)$ factors cancel, leaving the ratio of slopes.

        y
        |
        |         /  f(x)
        |        /
        |       /
        |      /
    ────┼─────●─────── x
        |    / a
        |   /  g(x)
        |  /
        | /

The Rule Extends To:

When to Apply Multiple Times

If after one application you still get $\frac{0}{0}$ or $\frac{\infty}{\infty}$, apply again:

$$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} = \lim_{x \to a} \frac{f''(x)}{g''(x)} = \cdots$$

But always verify the form before each application!

Worked Examples

Example 1: Basic $\frac{0}{0}$

Evaluate $\lim_{x \to 1} \frac{\ln x}{x - 1}$

Step 1: Verify the form

• Numerator: $\ln 1 = 0$ ✓
• Denominator: $1 - 1 = 0$ ✓
• Form: $\frac{0}{0}$ (L’Hospital applies)

Step 2: Differentiate top and bottom $$\lim_{x \to 1} \frac{\ln x}{x - 1} = \lim_{x \to 1} \frac{\frac{d}{dx}[\ln x]}{\frac{d}{dx}[x-1]} = \lim_{x \to 1} \frac{1/x}{1} = \lim_{x \to 1} \frac{1}{x} = 1$$

Example 2: Multiple Applications

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

Step 1: Verify the form

• Numerator: $e^\infty = \infty$ ✓
• Denominator: $\infty^2 = \infty$ ✓
• Form: $\frac{\infty}{\infty}$ (L’Hospital applies)

Step 2: First application $$\lim_{x \to \infty} \frac{e^x}{x^2} = \lim_{x \to \infty} \frac{e^x}{2x}$$

Step 3: Check again. Still $\frac{\infty}{\infty}$, so apply again: $$\lim_{x \to \infty} \frac{e^x}{2x} = \lim_{x \to \infty} \frac{e^x}{2} = \infty$$

Example 3: Simplify After Differentiating

Evaluate $\lim_{x \to \infty} \frac{\ln x}{\sqrt{x}}$

Step 1: Verify. Form is $\frac{\infty}{\infty}$ ✓

Step 2: Apply L’Hospital’s Rule $$\lim_{x \to \infty} \frac{\ln x}{\sqrt{x}} = \lim_{x \to \infty} \frac{1/x}{1/(2\sqrt{x})}$$

Step 3: Simplify instead of applying again $$= \lim_{x \to \infty} \frac{1/x}{1/(2\sqrt{x})} = \lim_{x \to \infty} \frac{2\sqrt{x}}{x} = \lim_{x \to \infty} \frac{2}{\sqrt{x}} = 0$$

Insight: Sometimes simplifying algebraically is faster than repeated L’Hospital applications.

Practice Problems

Common Misconceptions

Mastery Checklist

• I always verify the form is $\frac{0}{0}$ or $\frac{\infty}{\infty}$ BEFORE applying L’Hospital’s Rule
• I differentiate numerator and denominator SEPARATELY (not using the quotient rule)
• I can apply L’Hospital’s Rule multiple times when needed
• I recognize when simplifying is more efficient than repeated applications
• I know that L’Hospital fails if the limit of derivatives doesn’t exist
• I can use the rule for one-sided limits and limits at infinity

Mental Model

The “Zoom In” Analogy:

Near a point where both numerator and denominator equal zero, if you zoom in enough on their graphs, they look like straight lines through the origin.

The ratio of two lines through the origin is just the ratio of their slopes: $$\frac{\text{slope of } f}{\text{slope of } g} = \frac{f'(a)}{g'(a)}$$

L’Hospital’s Rule says: “For these troublesome $\frac{0}{0}$ limits, just compare the slopes instead!”

For $\frac{\infty}{\infty}$, think of it as a race to infinity. Taking derivatives reveals who’s growing faster.

Connections

Looking back:

• Recognizing Indeterminate Forms: must identify form FIRST
• Differentiation Rules: need to differentiate correctly

Looking ahead:

• L’Hospital’s Rule (Products & Differences): extends to $0 \cdot \infty$ and $\infty - \infty$
• L’Hospital’s Rule (Powers): extends to $0^0$, $\infty^0$, $1^\infty$
• Improper Integrals: often need L’Hospital for convergence analysis

Last updated: 2026-01-22