Mathematical Analysis I: Lecture 29

Mathematical Analysis I: Lecture 29

Lecturer: Yoh Tanimoto

09/11/2020

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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

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Bernoulli-de l¡¯Ho?pital 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 ¡Ê (a, b) such that

f 0 (y )(g(b) ? g(a)) = g 0 (y )(f (b) ? f (a)).

(Bernoulli-)de l¡¯Ho?pital rule is a useful tool to compute limits of the type

¡Þ

0

0 or ¡Þ .

Lecturer: Yoh Tanimoto

Mathematical Analysis I

09/11/2020

3 / 17

Bernoulli-de l¡¯Ho?pital rule

Theorem (Bernoulli-de l¡¯Ho?pital, case 1)

Let a < x0 , f , g differentiable in (a, x0 ) such that g 0 (x ) 6= 0 for x

sufficiently close to x0 , x 6= x0 , limx ¡úx ? f (x ) = limx ¡úx ? g(x ) = 0,

limx ¡úx ?

0

limx ¡úx ?

0

f 0 (x )

g 0 (x ) = L

f (x )

g(x ) = L.

0

0

¡Ê R. Then g(x ) 6= 0 for x close to x0 , x 6= x0 and

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¡¯Ho?pital. The limit limx ¡úx ?

0

0

)

limx ¡úx0 gf 0(x

(x ) .

Lecturer: Yoh Tanimoto

Mathematical Analysis I

f (x )

g(x )

is determined by

09/11/2020

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