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Guillaume De l'Hôpital 1661 - 1704 .1: L’Hôpital’s Rule Actually, L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. De l’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons. Greg Kelly, Hanford High School, Richland, Washin
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Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

Mar 26, 2015

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Page 1: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

Guillaume De l'Hôpital1661 - 1704

8.1: L’Hôpital’s Rule

Actually, L’Hôpital’s Rule was developed by his teacher Johann Bernoulli. De l’Hôpital paid Bernoulli for private lessons, and then published the first Calculus book based on those lessons.

Greg Kelly, Hanford High School, Richland, Washington

Page 2: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

Johann Bernoulli1667 - 1748

8.1: L’Hôpital’s Rule

Page 3: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

Zero divided by zero can not be evaluated, and is an example of indeterminate form.

2

2

4lim

2x

x

x

Consider:

If we try to evaluate this by direct substitution, we get:0

0

In this case, we can evaluate this limit by factoring and canceling:

2

2

4lim

2x

x

x

2

2 2lim

2x

x x

x

2lim 2x

x

4

Page 4: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

If we zoom in far enough, the curves will appear as straight lines.

2

2

4lim

2x

x

x

The limit is the ratio of the numerator over the denominator as x approaches 2.

2 4x

2x

limx a

f x

g x

Page 5: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

2

2

4lim

2x

x

x

limx a

f x

g x

f x

g x f x

g x

As 2x

becomes:

Page 6: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

2

2

4lim

2x

x

x

limx a

f x

g x

As 2x

f x

g xbecomes:

df

dg

df

dg

dx

dxdf

xdgd

Page 7: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

2

2

4lim

2x

x

x

limx a

f x

g x

2

2

4lim

2x

dx

dxdx

dx

2

2lim

1x

x

4

L’Hôpital’s Rule:

If is indeterminate, then:

limx a

f x

g x

lim limx a x a

f x f x

g x g x

Page 8: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

We can confirm L’Hôpital’s rule by working backwards, and using the definition of derivative:

f a

g a

lim

lim

x a

x a

f x f a

x ag x g a

x a

limx a

f x f a

x ag x g a

x a

limx a

f x f a

g x g a

0lim

0x a

f x

g x

limx a

f x

g x

Page 9: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

Example:

20

1 coslimx

x

x x

0

sinlim

1 2x

x

x

0

If it’s no longer indeterminate, then STOP!

If we try to continue with L’Hôpital’s rule:

0

sinlim

1 2x

x

x

0

coslim

2x

x

1

2 which is wrong,

wrong, wrong!

Page 10: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

On the other hand, you can apply L’Hôpital’s rule as many times as necessary as long as the fraction is still indeterminate:

20

1 12lim

x

xx

x

1

2

0

1 11

2 2lim2x

x

x

0

0

0

0

0

0not

1

2

20

11 1

2limx

x x

x

3

2

0

11

4lim2x

x

14

2

1

8

(Rewritten in exponential form.)

Page 11: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

L’Hôpital’s rule can be used to evaluate other indeterminate0

0forms besides .

The following are also considered indeterminate:

0 1 00 0

The first one, , can be evaluated just like .

0

0

The others must be changed to fractions first.

Page 12: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

1lim sinx

xx

This approaches0

0

1sin

lim1x

x

x

This approaches 0

We already know that0

sinlim 1x

x

x

but if we want to use L’Hôpital’s rule:

2

2

1 1cos

lim1x

x x

x

1sin

lim1x

x

x

1lim cosx x

cos 0 1

Page 13: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

1

1 1lim

ln 1x x x

If we find a common denominator and subtract, we get:

1

1 lnlim

1 lnx

x x

x x

Now it is in the form0

0

This is indeterminate form

1

11

lim1

lnx

xx

xx

L’Hôpital’s rule applied once.

1

1lim

ln 1x

x

x x

0

0Fractions cleared. Still

Page 14: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

1

1 1lim

ln 1x x x

1

1 lnlim

1 lnx

x x

x x

1

11

lim1

lnx

xx

xx

1

1lim

ln 1x

x

x x

1

1lim

1l

x

x

L’Hôpital again.

1

2

Page 15: Guillaume De l'Hôpital 1661 - 1704 8.1: LHôpitals Rule Actually, LHôpitals Rule was developed by his teacher Johann Bernoulli. De lHôpital paid Bernoulli.

Indeterminate Forms: 1 00 0

Evaluating these forms requires a mathematical trick to change the expression into a fraction.

lnlim lim lim f x L

x a x a x af x L f x e e

1/lim x

xx

1/lim ln x

xx

e

1lim lnx

xxe

lnlimx

x

xe

1

lim1x

x

e

0e 1

0

L’Hôpitalapplied