Overview

This topic covers how to resolve indeterminate forms using L’Hospital’s Rule and introduces higher-order derivatives, which appear in motion problems, Taylor series, and curvature analysis. Both expand your calculus toolkit beyond the basics.

Key Concepts and Structures

• L’Hospital’s Rule: If \lim_{x \to a} \frac{f(x)}{g(x)} gives 0/0 or ∞/∞, then
  \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} (if the latter limit exists)
• When to Use:
  • Only with 0/0 or ∞/∞ indeterminate forms
  • Must differentiate numerator and denominator separately
• Common Forms: \ln(x)/x, \sin(x)/x, e^x/x^n
• Higher-Order Derivatives:
  • f''(x) is the second derivative — rate of change of f'(x)
  • f^{(n)}(x) denotes the nth derivative
  • Second derivative is used in acceleration and concavity

Quick Tip

Practice Problems

1. Evaluate: \lim_{x \to 0} \frac{\sin x}{x}
   Show Solution

   Step 1: Direct substitution gives 0/0

   Step 2: Apply L’Hospital’s Rule: derivative of top is \cos x, of bottom is 1

   Step 3: Limit becomes \lim_{x \to 0} \cos x = 1

   Final Answer: 1

2. Find the third derivative of f(x) = x^4
   Show Solution

   f'(x) = 4x^3

   f''(x) = 12x^2

   f^{(3)}(x) = 24x

   Final Answer: f^{(3)}(x) = 24x