The Main Idea 

• Purpose:
  • Evaluates limits that are otherwise challenging to compute directly
  • Resolves indeterminate forms of type $\frac{0}{0}$ and $\frac{\infty}{\infty}$
• The Rule (0/0 case):
  • If $\lim_{x \to a} f(x) = 0$ and $\lim_{x \to a} g(x) = 0$, then: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$
• The Rule (∞/∞ case):
  • If $\lim_{x \to a} f(x) = \pm\infty$ and $\lim_{x \to a} g(x) = \pm\infty$, then: $\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$
• Applicability:
  • Works for one-sided limits
  • Applies when $a = \infty$ or $-\infty$
  • Can be applied repeatedly if needed
• Limitations:
  • Only applicable to indeterminate forms $\frac{0}{0}$ and $\frac{\infty}{\infty}$
  • Not applicable when limits of numerator and denominator are finite and non-zero

The Main Idea 

• Additional Indeterminate Forms:
  • $0 \cdot \infty$
  • $\infty - \infty$
  • $1^{\infty}$
  • $\infty^0$
  • $0^0$
• Strategy for L’Hôpital’s Rule:
  • Rewrite the expression to create a fraction of the form $\frac{0}{0}$ or $\frac{\infty}{\infty}$
  • Apply L’Hôpital’s Rule to the resulting fraction
• Handling $0 \cdot \infty$ Form:
  • Rewrite as a fraction, typically $\frac{\text{finite term}}{\frac{1}{\text{infinite term}}}$
• Handling $\infty - \infty$ Form:
  • Combine terms over a common denominator
• Handling Exponential Forms:
  • Use logarithms to convert to a product or quotient
  • Apply L’Hôpital’s Rule to the resulting expression

The Main Idea 

• Comparing Growth Rates:
  • Functions can approach infinity at different rates
  • We compare their relative growth as $x \to \infty$
• Defining Faster Growth:
  • $g(x)$ grows faster than $f(x)$ if: $\lim_{x \to \infty} \frac{g(x)}{f(x)} = \infty$ or $\lim_{x \to \infty} \frac{f(x)}{g(x)} = 0$
• Same Growth Rate:
  • $f(x)$ and $g(x)$ grow at the same rate if: $\lim_{x \to \infty} \frac{f(x)}{g(x)} = M$, where $M$ is a non-zero constant
• Hierarchy of Growth Rates:
  • Exponential > Power > Logarithmic
  • $e^x$ grows faster than $x^p$ for any $p > 0$
  • $x^p$ grows faster than $\ln x$ for any $p > 0$