109. L'Hôpital's Rule

Evaluating limits that result in a indeterminite form like or

If and (or both go to ), and and are differentiable near , then:

Example

Consider:

Step 1: Direct Substitution

Substituting :

Since this is an indeterminate form, we apply L’Hôpital’s Rule.

Step 2: First Application of L’Hôpital’s Rule

Differentiate the numerator and denominator:

  • Numerator:

  • Denominator:

Thus, applying L’Hôpital’s Rule:

Step 3: Second Check for Indeterminate Form

Substituting :

Since this is still an indeterminate form, we apply L’Hôpital’s Rule again.

Step 4: Second Application of L’Hôpital’s Rule

Differentiate again:

  • Numerator:

  • Denominator:

Applying L’Hôpital’s Rule again:

Step 5: Evaluate the Limit

Now, substituting :

Final Answer: