proof of l’Hôpital’s rule for ∞/∞ form

proof of l’Hôpital’s rule for

∞

/

∞

form

This is the proof of L’Hôpital’s Rule (

http://planetmath.org/LHpitalsRule

)
in the case of the

indeterminate form

±

∞

/

∞

.
Compared to the proof for the

0

/

0

case (

http://planetmath.org/ProofOfDeLHopitalsRule

),
more complicated estimates are needed.

Assume that

lim

x

→

a

⁡

f

⁢

(

x

)

=

±

∞

,

lim

x

→

a

⁡

g

⁢

(

x

)

=

±

∞

,

lim

x

→

a

⁡

f

′

⁢

(

x

)

g

′

⁢

(

x

)

=

m

,

where

a

and

m

are

real numbers

.
The case when

a

or

m

is

infinite

only involves
slight modifications to the

arguments

below.

Given

ϵ

>

0

.
there is a

δ

>

0

such that

|

f

′

⁢

(

ξ

)

g

′

⁢

(

ξ

)

-

m

|

<

ϵ

whenever

0

<

|

ξ

-

a

|

<

δ

.

Let

c

and

x

be points such that

a

-

δ

<

c

<

x

<

a

or

a

<

x

<

c

<

a

+

δ

.
(That is, both

c

and

x

are within

distance

δ

of

a

,
but

x

is always closer.)
By Cauchy’s mean value theorem, there exists some

ξ

x

in between

c

and

x

(and hence

0

<

|

ξ

x

-

a

|

<

δ

)
such that

f

⁢

(

x

)

-

f

⁢

(

c

)

g

⁢

(

x

)

-

g

⁢

(

c

)

=

f

′

⁢

(

ξ

x

)

g

′

⁢

(

ξ

x

)

.

We can assume the values

f

⁢

(

x

)

,

g

⁢

(

x

)

,

f

⁢

(

x

)

-

f

⁢

(

c

)

,

g

⁢

(

x

)

-

g

⁢

(

c

)

are

all non-zero

when

x

is close enough to

a

,
say, when

0

<

|

x

-

a

|

<

δ

′

for some

0

<

δ

′

<

δ

.
(So there is no

division by zero

in our equations.)
This is because

f

⁢

(

x

)

and

g

⁢

(

x

)

were assumed to approach

±

∞

,
so when

x

is close enough to

a

,
they will exceed the fixed values

f

⁢

(

c

)

,

g

⁢

(

c

)

, and

0

.

We write

f

⁢

(

x

)

g

⁢

(

x

)

=

f

⁢

(

x

)

f

⁢

(

x

)

-

f

⁢

(

c

)

⋅

g

⁢

(

x

)

-

g

⁢

(

c

)

g

⁢

(

x

)

⋅

f

⁢

(

x

)

-

f

⁢

(

c

)

g

⁢

(

x

)

-

g

⁢

(

c

)

=

1

-

g

⁢

(

c

)

/

g

⁢

(

x

)

1

-

f

⁢

(

c

)

/

f

⁢

(

x

)

⋅

f

′

⁢

(

ξ

x

)

g

′

⁢

(

ξ

x

)

.

Note that

lim

x

→

a

⁡

1

-

g

⁢

(

c

)

/

g

⁢

(

x

)

1

-

f

⁢

(

c

)

/

f

⁢

(

x

)

=

1

,

but

ξ

x

is not guaranteed to approach

a

as

x

approaches

a

,
so we cannot just take the limit

x

→

a

directly. However:
there exists

0

<

δ

′′

<

δ

′

so that

|

1

-

g

⁢

(

c

)

/

g

⁢

(

x

)

1

-

f

⁢

(

c

)

/

f

⁢

(

x

)

-

1

|

<

ϵ

|

m

|

+

ϵ

whenever

0

<

|

x

-

a

|

<

δ

′′

.
Then

|

f

⁢

(

x

)

g

⁢

(

x

)

-

m

|

=

|

(

f

′

⁢

(

ξ

x

)

g

′

⁢

(

ξ

x

)

-

m

)

+

f

′

⁢

(

ξ

x

)

g

′

⁢

(

ξ

x

)

⁢

(

1

-

g

⁢

(

c

)

/

g

⁢

(

x

)

1

-

f

⁢

(

c

)

/

f

⁢

(

x

)

-

1

)

|

≤

ϵ

+

(

|

m

|

+

ϵ

)

⁢

ϵ

|

m

|

+

ϵ

=

2

⁢

ϵ

for

0

<

|

x

-

a

|

<

δ

′′

.

This proves

lim

x

→

a

⁡

f

⁢

(

x

)

g

⁢

(

x

)

=

m

=

lim

x

→

a

⁡

f

′

⁢

(

x

)

g

′

⁢

(

x

)

.

References

1

Michael Spivak,

Calculus

, 3rd ed. Publish or Perish, 1994.

Title

proof of l’Hôpital’s rule for

∞

/

∞

form

Canonical name

ProofOfLHopitalsRuleForinftyinftyForm

Date of creation

2013-03-22 15:40:15

Last modified on

2013-03-22 15:40:15

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

7

Author

stevecheng (10074)

Entry type

Proof

Classification

msc 26A06

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