Further Limits (DP IB Analysis & Approaches (AA)): Revision Note

l'Hôpital's Rule

What is l’Hôpital’s Rule?

• l’Hôpital’s rule is a method involving calculus that allows us to find the value of certain limits

• Specifically, it allows us to attempt to evaluate the limit of a quotient  for which our usual limit evaluation techniques would return one of the indeterminate forms  or .

How do I evaluate a limit using l’Hôpital’s Rule?

• STEP 1: Check that the limit of the quotient results in one of the indeterminate forms given above

  • I.e., check that or  

• STEP 2: Find the derivatives of the numerator and denominator of the quotient

• STEP 3: Check whether the limit  exists

• STEP 4: If that limit does exist, then 

• STEP 5: If or  then you may repeat the process by considering  (and possibly higher order derivatives after that)

  • As long as the limits continue giving indeterminate forms you may continue applying l’Hôpital’s rule

  • Each time this happens find the next set of derivatives and consider the limit again

Limits Using a Maclaurin Series

How do I evaluate a limit using Maclaurin series?

• Limits of the form  or  may sometimes be evaluated by using Maclaurin series

• Usually this will be in a situation where attempting to evaluate the limit in the usual way returns an indeterminate form  or .

• In such a case:

  • STEP 1: Find the Maclaurin series for  and

  • STEP 2: Rewrite  using the Maclaurin series in the numerator and denominator

  • STEP 3: Use algebra to simplify your new expression for  as far as possible

  • STEP 4: Evaluate the limit using your simplified form of the expression