L'Hospital's Rule and Indeterminate Forms

L'Hospital's Rule

Introduction to L'Hospital's Rule

L'Hospital's Rule is a powerful tool in calculus for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. This rule allows us to differentiate the numerator and denominator separately to find the limit.

• Indeterminate Forms: Expressions like 0/0 or ∞/∞ that do not have a clear value and require further analysis.

• Applicability: L'Hospital's Rule can only be used when the original limit yields an indeterminate form.

Statement of L'Hospital's Rule

If limx→a f(x)/g(x) yields an indeterminate form 0/0 or ∞/∞, then:

$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}$

• f(x) and g(x) must be differentiable near x = a.

• g'(x) \neq 0 near x = a (except possibly at x = a).

• The limit on the right side must exist or be infinite.

Derivation and Explanation

Suppose f(x) and g(x) are differentiable at x = a and both approach 0 as x → a. Using the definition of the derivative, we can write:

$\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f(a) + f'(a)(x-a)}{g(a) + g'(a)(x-a)}$

Since f(a) = 0 and g(a) = 0, this simplifies to:

$\lim_{x \to a} \frac{f'(a)(x-a)}{g'(a)(x-a)} = \frac{f'(a)}{g'(a)}$

This justifies the use of derivatives in evaluating the original limit.

Examples of L'Hospital's Rule

• Example 1: $\lim_{x \to 0} \frac{1 - e^{-2x}}{x}$

  • Direct substitution gives 0/0 (indeterminate form).

  • Apply L'Hospital's Rule:

  • $\lim_{x \to 0} \frac{2e^{-2x}}{1} = 2e^{0} = 2$

• Example 2: $\lim_{x \to 0} e^{-2x}$

  • Direct substitution gives $e^{0} = 1$ (not indeterminate).

  • No need for L'Hospital's Rule.

• Example 3: $\lim_{x \to 0} x \ln|x|$

  • Direct substitution gives $0 \cdot (-\infty)$ (indeterminate form).

  • Rewrite as $\lim_{x \to 0} \frac{\ln|x|}{1/x}$ (ratio form).

  • Apply L'Hospital's Rule:

  • $\lim_{x \to 0} \frac{1/x}{-1/x^2} = \lim_{x \to 0} -x = 0$

Summary Table: Indeterminate Forms and L'Hospital's Rule

Form	Indeterminate?	Can Use L'Hospital's Rule?
0/0	Yes	Yes
∞/∞	Yes	Yes
0 × ∞	Yes (convert to 0/0 or ∞/∞)	Yes (after conversion)
1/∞	No	No

Key Points to Remember

• Always check if the limit is indeterminate before applying L'Hospital's Rule.

• Rewrite limits into a ratio form if necessary (e.g., $x \ln x$ to $\frac{\ln x}{1/x}$).

• Differentiate numerator and denominator separately.

• Repeat the process if the result is still indeterminate.