l'Hôpital's Rule (DP IB Analysis & Approaches (AA): HL): Revision Note

l'Hôpital's rule

What is l’Hôpital’s Rule?

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

• Specifically, it allows you to attempt to evaluate the limit of a quotient $\frac{f\left(x\right)}{g\left(x\right)}$ for which the usual limit evaluation techniques would return one of the indeterminate forms $\frac{0}{0}$ or $\frac{\pm \infty }{\pm \infty }$.

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 $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{f\left(a\right)}{g\left(a\right)}=\frac{0}{0}$ or $\frac{\pm \infty }{\pm \infty }$ 

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

• STEP 3
  Check whether the limit $\underset{x\to a}{\lim }\frac{f^{\text{'}}\left(x\right)}{g^{\text{'}}\left(x\right)}$ exists
   

• STEP 4
  If that limit does exist, then $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to a}{\lim }\frac{f^{\text{'}}\left(x\right)}{g^{\text{'}}\left(x\right)}$

• STEP 5
  If $\underset{x\to a}{\lim }\frac{f^{\text{'}}\left(x\right)}{g^{\text{'}}\left(x\right)}=\frac{f^{\text{'}}\left(a\right)}{g^{\text{'}}\left(a\right)}=\frac{0}{0}$or $\frac{\pm \infty }{\pm \infty }$ then you may repeat the process by considering $\underset{x\to a}{\lim }\frac{f^{\text{'}\text{'}}\left(x\right)}{g^{\text{'}\text{'}}\left(x\right)}$ (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