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mcastillo356

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- TL;DR Summary
- Bernouili's work, french mathematician's work, l'Hôpital first rule...Need some more knowledge

Hi, PF

Got questions to start with: ¿some casual background about these Rules?; ¿are them two, as the textbook says?.

https://en.wikipedia.org/wiki/L'Hôpital's_rule (only one statement found)

Here goes the first, from "Calculus, 7th ed, R, Adams, C. Essex"

THEOREM 3 The first l'Hôpital Rule

Suppose the functions ##f## and ##g## are differentiable on the interval ##(a,b)## and ##g'(x)\neq 0## there. Suppose also that

(i) ##\lim_{x\to a^+}f(x)=\lim{x\to a^+}g(x)=0## and

(ii) ##\lim_{x\to a^+}{\frac{f′(x)}{g′(x)}}=L## (where ##L## is finite or ##\infty## or ##−\infty##)

Then

##\lim_{x\to a^+}{\frac{f(x)}{g(x)}=L}##

Similar results hold if every occurrence of ##\lim_{x\rightarrow {a^+}}## is replaced by ##\lim_{x\rightarrow {b^-}}## or even ##\lim_{x\rightarrow{c^+}}## where ##a<c<b##. The cases ##a=-\infty## and ##b=\infty## are also allowed

PROOF We prove the case involving ##\lim_{x\rightarrow{a^+}}## for finite ##a##. Define

##F(x)=\begin{cases}f(x)&\mbox{if}a<x<b\\0&\mbox{if} x=a \end{cases}##

and

##G(x)=\begin{cases}g(x)&\mbox{if}a<x<b\\0&\mbox{if} x=a \end{cases}##

Then ##F## and ##G## are continuous on the interval ##[a,x]## and differentiable on the interval ##(a,x)## for every ##x## in ##(a,b)##. By the Generalized Mean-Value Theorem (...) there exists a number ##c## in ##(a,x)## such that

##\frac{f(x)}{g(x)}=\frac{F(x)}{G(x)}=\frac{F(x)-F(a)}{G(x)-G(a)}=\frac{F'(c)}{G'(c)}=\frac{f'(c)}{g'(c)}##.

Since ##a<c<x##, if ##x\rightarrow{a^+}##, then neccesarily ##c\rightarrow{a^+}##, so we have

##\lim{x\to{a^+}}{\frac{f(x)}{g(x)}}=\lim{c\to{a^+}}{\frac{f'(c)}{g'(c)}}=L##

Mean Value Theorem seems a limitless tool in Analysis. Question: ##\lim{c\to{a^+}}{\frac{f'(c)}{g'(c)}}=L=\frac{f'(c)}{g'(c)}##? Think so. At this point, ##c\rightarrow{a^+}## doesn´t add worth information; it's a useless limit

Attemtp: Wikipedia isn't wrong; is straight, I guess; but incomprehensive for me. I understand what the textbook says, but need some kind of text comment on the aim of my textbook.Thanks. I think LaTeX is not well done, please PF, check it.

Edited at 6:39 AM Europe timing

Got questions to start with: ¿some casual background about these Rules?; ¿are them two, as the textbook says?.

https://en.wikipedia.org/wiki/L'Hôpital's_rule (only one statement found)

Here goes the first, from "Calculus, 7th ed, R, Adams, C. Essex"

THEOREM 3 The first l'Hôpital Rule

Suppose the functions ##f## and ##g## are differentiable on the interval ##(a,b)## and ##g'(x)\neq 0## there. Suppose also that

(i) ##\lim_{x\to a^+}f(x)=\lim{x\to a^+}g(x)=0## and

(ii) ##\lim_{x\to a^+}{\frac{f′(x)}{g′(x)}}=L## (where ##L## is finite or ##\infty## or ##−\infty##)

Then

##\lim_{x\to a^+}{\frac{f(x)}{g(x)}=L}##

Similar results hold if every occurrence of ##\lim_{x\rightarrow {a^+}}## is replaced by ##\lim_{x\rightarrow {b^-}}## or even ##\lim_{x\rightarrow{c^+}}## where ##a<c<b##. The cases ##a=-\infty## and ##b=\infty## are also allowed

PROOF We prove the case involving ##\lim_{x\rightarrow{a^+}}## for finite ##a##. Define

##F(x)=\begin{cases}f(x)&\mbox{if}a<x<b\\0&\mbox{if} x=a \end{cases}##

and

##G(x)=\begin{cases}g(x)&\mbox{if}a<x<b\\0&\mbox{if} x=a \end{cases}##

Then ##F## and ##G## are continuous on the interval ##[a,x]## and differentiable on the interval ##(a,x)## for every ##x## in ##(a,b)##. By the Generalized Mean-Value Theorem (...) there exists a number ##c## in ##(a,x)## such that

##\frac{f(x)}{g(x)}=\frac{F(x)}{G(x)}=\frac{F(x)-F(a)}{G(x)-G(a)}=\frac{F'(c)}{G'(c)}=\frac{f'(c)}{g'(c)}##.

Since ##a<c<x##, if ##x\rightarrow{a^+}##, then neccesarily ##c\rightarrow{a^+}##, so we have

##\lim{x\to{a^+}}{\frac{f(x)}{g(x)}}=\lim{c\to{a^+}}{\frac{f'(c)}{g'(c)}}=L##

Mean Value Theorem seems a limitless tool in Analysis. Question: ##\lim{c\to{a^+}}{\frac{f'(c)}{g'(c)}}=L=\frac{f'(c)}{g'(c)}##? Think so. At this point, ##c\rightarrow{a^+}## doesn´t add worth information; it's a useless limit

Attemtp: Wikipedia isn't wrong; is straight, I guess; but incomprehensive for me. I understand what the textbook says, but need some kind of text comment on the aim of my textbook.Thanks. I think LaTeX is not well done, please PF, check it.

Edited at 6:39 AM Europe timing

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