L'Hospital's Rule (College Board AP® Calculus AB)

Indeterminate forms

What is an indeterminate form?

• An indeterminate form is a mathematical expression of one of the two following forms:

  • 

  • 

• The value of an indeterminate form is undefined

  • Dividing by 0 always gives an undefined expression

  • And note that, for example, is not equal to 1

    • is not a number

    • so it can't be canceled to simplify a fraction

• Sometimes attempting to evaluate a limit using substitution leads to one of the indeterminate forms given above

  • L'Hospital's rule provides a method for dealing with limits of that form

Evaluating limits using L'Hospital's rule

What is L'Hospital’s Rule?

• L'Hospital's rule (sometimes written as L’Hôpital’s rule) is a method for finding the value of certain limits using calculus

  • Specifically, it allows us to attempt to evaluate the limit of a quotient  

  • for which attempting to evaluate the limit by substitution returns one of the indeterminate forms  or

• For such a quotient function, L'Hospital's rule says that

  • 

  • I.e., you can take the derivatives of the numerator and denominator

    • and attempt to evaluate the limit again in that form

How do I evaluate a limit using L’Hospital’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’Hospital’s rule

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