Notes on Using L'Hôpital's Rule to Evaluate Limits

Introduction

• Discussing the evaluation of a limit using L'Hôpital's Rule.
• Example: ( \lim_{{x \to \infty}} \left(1 + \frac{3}{x}\right)^x )
• Expression as ( x \to \infty ):
  • ( \frac{3}{x} \to 0 )
  • Base of the expression goes to 1.
  • Exponent goes to infinity.
• Form: "1 to the infinity" (indeterminate)._

Analysis of Indeterminate Form

• Base approaching 1 suggests result may approach 1.
• Exponent increasing suggests result may approach infinity.
• Truth lies in between.

Using L'Hôpital's Rule

• Call the limit ( L ).
• Calculate ( \ln{L} ).
• This approach turns the expression into a form where L'Hôpital's Rule applies.
• Logarithms convert the expression into a product form.

Steps in Applying L'Hôpital's Rule

1. Taking Logarithm:
   • Convert expression to ( x \cdot \ln{(1 + \frac{3}{x})} ).
   • Analyze form: Infinity * 0 (still indeterminate).
2. Convert to Fraction:
   • Rewrite as ( \frac{\ln{(1 + \frac{3}{x})}}{\frac{1}{x}} ).
   • Form: 0/0, suitable for L'Hôpital's Rule.
3. Apply L'Hôpital's Rule:
   • Differentiate numerator and denominator.
   • Use chain rule for numerator.
   • Result: ( \frac{3}{1 + \frac{3}{x}} ).
   • Limit becomes ( \frac{3}{1} = 3 ).
4. Exponentiate Result:
   • ( L = e^3 ).*

Evaluating Limit as ( x \to 0^+ )

• Re-evaluate for ( \lim_{{x \to 0^+}} \left(1 + \frac{3}{x}\right)^x ).
• Form: Infinity to the zero (indeterminate).
• Same procedure:
  • Logarithm approach.
  • Convert to product ( 0 \cdot \infty ).
  • Convert to fraction. Apply L'Hôpital's Rule again.
  • Result: Approaches 0.
  • ( L = e^0 = 1 )._

Graphical Representation

• Graph interpretation:
  • ( x \to \infty ): Function approaches ( e^3 ).
  • ( x \to 0^+ ): Function approaches 1.
  • Discontinuity at ( x = 0 ) (hole in the graph).

Conclusion

• L'Hôpital's Rule effectively resolves the indeterminate forms and calculates limits.
• Understanding of form analysis and transformations is crucial in applying L'Hôpital's Rule effectively.