4.7 - L’Hopital’s Rule; 4.7 - L’Hopital’s Rule; Indeterminate FormsIndeterminate Forms

(page 277-284)(page 277-284)

L’Hopital’s Rule was developed by and L’Hopital’s Rule was developed by and named after a French mathematician - named after a French mathematician - Guillame Francois Antonine De L’Hopital Guillame Francois Antonine De L’Hopital - (1661-1704).- (1661-1704).

L’Hopital’s Rule is used to find limits of L’Hopital’s Rule is used to find limits of rational expressions whose numerator rational expressions whose numerator and denominator both are zero.and denominator both are zero.

The formal development of L’Hopital’s The formal development of L’Hopital’s Rule will be done in Calculus B,CRule will be done in Calculus B,C

L’Hopital’s RuleL’Hopital’s Rule

In a rational expression, when the limit of the In a rational expression, when the limit of the numerator approaches zero, the ratio numerator approaches zero, the ratio approaches zero. When the limit of the approaches zero. When the limit of the denominator approaches zero the ratio denominator approaches zero the ratio approaches positive or negative infinity.approaches positive or negative infinity.

In the limit below, the two limits “off set” each In the limit below, the two limits “off set” each other and the limit is 1, as we discovered using other and the limit is 1, as we discovered using the squeeze theorem earlier in the coursethe squeeze theorem earlier in the course

0

sinlim 1x

x

x

Limits Using L’Hopital’s Limits Using L’Hopital’s RuleRule

lim limf x f x

g x g x

L’Hopital’s Rule simply stated is:

L’Hopital’s Rule AppliedL’Hopital’s Rule Applied

So, the limit of 0

sinlim 1x

x

x

Can be found easily as follows:

0

sinlimx

x

x

0

sinlimx

dx

dxdx

dx

11

10

0

lim cos

lim1x

x

x

L’Hopital’s Rule AppliedL’Hopital’s Rule Applied

0

1 coslimx

x

x

1 coslimx o

dx

dxdx

dx

sinlim

1x o

x

sin 0 0

When to use L’Hopital’s When to use L’Hopital’s RuleRule

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

0

6 6lim 3

2 2x

x

x

0

6lim

2x

x

x

0

6lim

2x

dx

dxdx

dx

0

1lim 1

1x

Calculus A,B and Calculus Calculus A,B and Calculus B,CB,C

L’Hopital’s Rule WILL NOT appear on L’Hopital’s Rule WILL NOT appear on the Calculus A,B Exam. It is on the the Calculus A,B Exam. It is on the Calculus B,C Exam.Calculus B,C Exam.

We will only do the simpler forms of We will only do the simpler forms of L’Hopital’s Rule for homework and L’Hopital’s Rule for homework and L’Hopital’s Rule will not be tested on in L’Hopital’s Rule will not be tested on in this course.this course.

There are otherThere are other indeterminate forms for indeterminate forms for which L’Hopital’s Rule can be used.which L’Hopital’s Rule can be used.

Other Indeterminate Other Indeterminate FormsForms

0Indeterminate Forms of Type

0

Indeterminate Forms of Type 0

Indeterminate Forms of Type

Indeterminate Forms of Type -

0 0Indeterminate Forms of Type 0 , ,1

L’Hopital’s Rule and L’Hopital’s Rule and Exponential GrowthExponential Growth

The above graphs suggest the following limits:

lim 0n

xx

x

e lim

x

nx

e

x