4.7 - L’Hopital’s Rule; Indeterminate Forms (page ) b L’Hopital’s Rule was developed by and named after a French mathematician - Guillame Francois Antonine De L’Hopital - ( ). b L’Hopital’s Rule is used to find limits of rational expressions whose numerator and denominator both are zero. b The formal development of L’Hopital’s Rule will be done in Calculus B,C

L’Hopital’s Rule b In a rational expression, when the limit of the numerator approaches zero, the ratio approaches zero. When the limit of the denominator approaches zero the ratio approaches positive or negative infinity. b In the limit below, the two limits “off set” each other and the limit is 1, as we discovered using the squeeze theorem earlier in the course

Limits Using L’Hopital’s Rule L’Hopital’s Rule simply stated is:

L’Hopital’s Rule Applied So, the limit of Can be found easily as follows:

L’Hopital’s Rule Applied

When to use L’Hopital’s Rule L’Hopital’s Rule only works for limits of rational expressions whose numerator and denominator result in the ratio zero/zero. Notice the following incorrect use of L’Hopital’s Rule.

Calculus A,B and Calculus B,C b L’Hopital’s Rule WILL NOT appear on the Calculus A,B Exam. It is on the Calculus B,C Exam. b We will only do the simpler forms of L’Hopital’s Rule for homework and L’Hopital’s Rule will not be tested on in this course. b There are other indeterminate forms for which L’Hopital’s Rule can be used.

Other Indeterminate Forms

L’Hopital’s Rule and Exponential Growth The above graphs suggest the following limits: