Mathematical Analysis I: Lecture 29

Mathematical Analysis I: Lecture 29

Lecturer: Yoh Tanimoto

09/11/2020 Start recording...

Annoucements

Tutoring (by Mr. Lorenzo Panebianco): Tuesday 15:00-16:30 until 10th November. Then move to Tuesday morning. Today: Apostol Vol. 1, Chapter 7.14.

Lecturer: Yoh Tanimoto

Mathematical Analysis I

09/11/2020 2 / 17

Bernoulli-de l'H^opital rule

Let us recall the mean value theorem of Cauchy: let f , g be continuous in [a, b] and differentiable in (a, b). Then there is y (a, b) such that

f (y )(g(b) - g(a)) = g (y )(f (b) - f (a)).

(Bernoulli-)de l'H^opital rule is a useful tool to compute limits of the type

0 0

or

.

Lecturer: Yoh Tanimoto

Mathematical Analysis I

09/11/2020 3 / 17

Bernoulli-de l'H^opital rule

Theorem (Bernoulli-de l'H^opital, case 1)

Let a < x0, f , g differentiable in (a, x0) such that g (x ) = 0 for x

sufficiently close to x0, x = x0, limxx0- f (x ) = limxx0- g (x ) = 0,

limx x0-

f g

(x ) (x )

=

L

R.

Then

g(x )

=

0

for

x

close

to

x0, x

=

x0

and

limx x0-

f (x ) g(x )

=

L.

A similar result holds for right limits.

Lecturer: Yoh Tanimoto

Mathematical Analysis I

09/11/2020 4 / 17

a(

( b

x0

Figure:

Theorem

of

de

l'H^opital.

The

limit

limx x0-

f (x ) g(x )

is

determined

by

limx x0

f g

(x ) (x )

.

Lecturer: Yoh Tanimoto

Mathematical Analysis I

09/11/2020 5 / 17

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