Lecture Notes: L'Hopital's Rule Overview

Introduction to L'Hopital's Rule

• Purpose: L'Hopital's Rule helps solve limits questions in calculus, specifically when answers aren't straightforward.
• Topic Context: Part of calculus, particularly useful in Topic 5.
• When to Use: Necessary when a limit question results in indeterminate forms.

Examples of Limit Problems

Example 1

• Problem: Limit as (x \to 0) of (\frac{\sin(x)}{x})
• Direct Substitution: Leads to (\frac{0}{0})

Example 2

• Problem: Limit as (x \to \infty) of (\frac{\ln(x)}{x})
• Direct Substitution: Results in (\frac{\infty}{\infty})

Example 3

• Problem: Limit as (x \to \infty) of (x \cdot e^{-x})
• Direct Substitution: Results in (\infty \cdot 0)

Indeterminate Forms

• Forms: (\frac{0}{0}), (\frac{\infty}{\infty}), (\infty \cdot 0)
• Resolution: Use L'Hopital's Rule to resolve these forms.

Applying L'Hopital's Rule

• Rule: If a limit results in an indeterminate form, take the derivatives of the numerator and the denominator.
• Process: Continue to take derivatives until the limit can be resolved.

Solving Example 1 with L'Hopital's Rule

• Derivatives:
  • Numerator: Derivative of (\sin(x)) is (\cos(x))
  • Denominator: Derivative of (x) is 1
• Result: Substitute (x = 0) yields (\frac{1}{1} = 1)

Solving Example 2 with L'Hopital's Rule

• Derivatives:
  • Numerator: Derivative of (\ln(x)) is (\frac{1}{x})
  • Denominator: Derivative of (x) is 1
• Result: Substitute (x = \infty) yields (\frac{0}{1} = 0)

Solving Example 3 with L'Hopital's Rule

• Reformulate: Rewrite as (\frac{x}{e^x})
• Derivatives:
  • Numerator: Derivative of (x) is 1
  • Denominator: Derivative of (e^x) is (e^x)
• Result: Substitute (x = \infty) yields (\frac{1}{\infty} = 0)

Conclusion

• General Rule: For the limit as (x) approaches some value (a), if direct substitution fails and both functions are differentiable, use derivatives to find the limit.
• Multiple Applications: May need multiple applications of L'Hopital's Rule if first derivatives still result in indeterminate forms.

Historical Note

• Discovery: L'Hopital's Rule was discovered by Johann Bernoulli, but named after L'Hopital who purchased the rights.