L’Hôpital’s Rule: Fresh Take

  • Spot indeterminate forms like in calculations, and use L’Hôpital’s rule to find precise values
  • Explain how quickly different functions increase or decrease compared to each other

L’Hôpital’s Rule

The Main Idea 

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

Evaluate [latex]\underset{x\to 0}{\lim}\dfrac{x}{\tan x}[/latex].

Evaluate [latex]\underset{x\to \infty }{\lim}\dfrac{\ln x}{5x}[/latex]

Other Indeterminate Forms

The Main Idea 

  • Additional Indeterminate Forms:
    • [latex]0 \cdot \infty[/latex]
    • [latex]\infty - \infty[/latex]
    • [latex]1^{\infty}[/latex]
    • [latex]\infty^0[/latex]
    • [latex]0^0[/latex]
  • Strategy for L’Hôpital’s Rule:
    • Rewrite the expression to create a fraction of the form [latex]\frac{0}{0}[/latex] or [latex]\frac{\infty}{\infty}[/latex]
    • Apply L’Hôpital’s Rule to the resulting fraction
  • Handling [latex]0 \cdot \infty[/latex] Form:
    • Rewrite as a fraction, typically [latex]\frac{\text{finite term}}{\frac{1}{\text{infinite term}}}[/latex]
  • Handling [latex]\infty - \infty[/latex] 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

Evaluate [latex]\underset{x\to \infty}{\lim} x^{\frac{1}{\ln x}}[/latex]

Evaluate [latex]\underset{x\to 0^+}{\lim} x^x[/latex]

Growth Rates of Functions

The Main Idea 

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

Compare the growth rates of [latex]x^{100}[/latex] and [latex]2^x[/latex].