Indeterminate Forms and L'Hôpital's Rule

CORE CONCEPT: L'HOPITAL'S RULE
L'Hopital's Rule is a method used to find the limit of a fraction when direct substitution results in an indeterminate form, specifically 0/0 or infinity/infinity. Instead of evaluating the original function, you take the derivative of the numerator and the derivative of the denominator separately and then try the limit again.

FORMULA:
Limit of [f(x) / g(x)] = Limit of [f'(x) / g'(x)]

IMPORTANT RULES:

  1. This is NOT the Quotient Rule. Do not take the derivative of the whole fraction. Differentiate the top and bottom individually.

  2. You can apply the rule multiple times. If the first derivatives still result in 0/0 or infinity/infinity, take the second derivatives.

INDETERMINATE FORMS:

  • Basic: 0/0 and (infinity / infinity).

  • Products: 0 times infinity (Rewrite as a fraction to use the rule).

  • Differences: infinity minus infinity (Find a common denominator to combine into one fraction).

  • Powers: 1 to the infinity, 0 to the 0, and infinity to the 0.

HOW TO SOLVE INDETERMINATE POWERS:

  1. Set the limit equal to "y".

  2. Take the natural log (ln) of both sides.

  3. Use log properties to move the exponent in front of the log.

  4. Solve the limit of the new expression.

  5. Since you solved for ln(y), your final answer is e raised to the power of whatever result you got.