Some questions on l'Hôpital rules

In summary: I've learnt today.In summary, L'Hôpital's rule is a theorem that states under certain conditions, the limit of a quotient of two functions is equal to the limit of their derivatives. This rule only applies when the limit is of the form 0/0 or ∞/∞. The theorem has been proven and shown to be accurate, but there are some differences in notation and understanding among different sources. The Mean Value Theorem is also a useful tool when proving L'Hôpital's rule, but it is important to note that the variable c in the theorem is not a constant, but a function of x.
  • #1
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
 
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  • #2
What's the question?
 
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  • #3
Hi, the questions are: is the limit when ##c\rightarrow{a^+}## of ##\dfrac{f'(c)}{g'(c)}## the same as the quotient ##\dfrac{f'(c)}{g'(c)}## itself? I think so because ##a<c##;
Wikipedia truly states only one l'Hôpital Rule? (the article, I mean, the maths at the second paragraph are fine: no need to mention two Rules -fine to me-);
Which source must I pick? I am tempted not to reject anyone; but doubts arise. Let's take 1st l'Hôpital rule: why does look at ##x## when tends to ##a## from the right?; Wikipedia talks about an interval ##I## an ##x## tending, presumably -but not for sure- to ##c## (more intuitive to me).
 
  • #4
mcastillo356 said:
Hi, the questions are: is the limit when ##c\rightarrow{a^+}## of ##\dfrac{f'(c)}{g'(c)}## the same as the quotient ##\dfrac{f'(c)}{g'(c)}## itself?
Only if the quotient is well defined at ##c##. It might be another limit of the form ##\frac 0 0##, for example.
 
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  • #5
PeroK said:
Only if the quotient is well defined at ##c##. It might be another limit of the form ##\frac 0 0##, for example.
Or ##\pm \frac \infty \infty##.
 
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  • #6
PeroK said:
Only if the quotient is well defined at ##c##. It might be another limit of the form ##\frac 0 0##, for example.
Could explain some more about this quote? No attempt, clue about it. The question is: how the quotient, and why, should be well defined at ##c#?. What means to be well defined? Don't manage well in Wikipedia.
 
  • #7
mcastillo356 said:
Could explain some more about this quote? No attempt, clue about it. The question is: how the quotient, and why, should be well defined at ##c#?. What means to be well defined? Don't manage well in Wikipedia.
To take an example: suppose we want to evaluate the following limit using L'Hopital's rule:
$$\lim_{x \to 0} \frac{x^3}{x^2}$$If we differentiate top and bottom, we get another indeterminate form:
$$\lim_{x \to 0} \frac{x^3}{x^2} =\lim_{x \to 0} \frac{3x^2}{2x}$$Your question, if I understand it, is that why don't we drop the limit on the RHS and evaluate the function at ##x = 0##? The answer is because we can't as we have another expression of the ##\frac 0 0##.

But, we can apply L'Hopital's rule a second time to get:
$$\lim_{x \to 0} \frac{3x^2}{2x} =\lim_{x \to 0} \frac{6x}{2} = 0$$This time we have got a function where the limit can be evaluated simply.

In general, you sometimes have to apply the L'Hopital rule several times.
 
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  • #8
mcastillo356 said:
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
But [itex]c[/itex] is not a constant: it is a function of [itex]x[/itex] as it is the result of applying the mean value theorem to a function on [itex][a,x][/itex]. So it is more accurate, and perhaps clearer, to write [tex]
\lim_{x \to a^{+}} \frac{f(x)}{g(x)} = \lim_{x \to a^{+}} \frac{f'(c(x))}{g'(c(x))}.[/tex] At this point we can replace [itex]c(x)[/itex] with a dummy variable (either introducing a new symbol or repurposing [itex]c[/itex] or [itex]x[/itex]) to get [tex]
\lim_{x \to a^{+}} \frac{f(x)}{g(x)} = \lim_{c \to a^{+}} \frac{f'(c)}{g'(c)}.[/tex]
 
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  • #9
PeroK said:
Your question, if I understand it, is that why don't we drop the limit on the RHS and evaluate the function at ##x = 0##? The answer is because we can't as we have another expression of the ##\frac 0 0##.
Well, if I've understood, right hand side limit and left hand side limit must be equal to say we have a limit when the argument tends to some ##a\in{\mathbb{R}}##. Basically they can differ in many basic limit operations.
 
  • #10
pasmith said:
But [itex]c[/itex] is not a constant: it is a function of [itex]x[/itex] as it is the result of applying the mean value theorem to a function on [itex][a,x][/itex]
Brilliant, thanks indeed for the post, I really thought I was facing reals. Nice remark
 

Related to Some questions on l'Hôpital rules

1. What is l'Hôpital's rule?

l'Hôpital's rule is a mathematical tool used to evaluate limits involving indeterminate forms, such as 0/0 or ∞/∞. It states that for certain functions, the limit of the quotient of their derivatives is equal to the limit of the original function.

2. When should l'Hôpital's rule be used?

l'Hôpital's rule should be used when evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It is also useful when evaluating limits involving trigonometric functions or exponential functions.

3. What are the conditions for applying l'Hôpital's rule?

The conditions for applying l'Hôpital's rule are that the limit must be in an indeterminate form, the functions must be differentiable in a neighborhood of the limit point, and the limit of the quotient of their derivatives must exist.

4. Are there any limitations to using l'Hôpital's rule?

Yes, there are limitations to using l'Hôpital's rule. It cannot be used to evaluate limits at infinity, and it may not work for some functions that do not satisfy the conditions for its application.

5. Can l'Hôpital's rule be applied multiple times?

Yes, l'Hôpital's rule can be applied multiple times as long as the resulting limit is still in an indeterminate form and the conditions for its application are met. However, it is important to be cautious and check for other possible methods of evaluation before using l'Hôpital's rule repeatedly.

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