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 and are differentiable on the interval and there. Suppose also that
(i) and
(ii) (where is finite or or )
Then

Similar results hold if every occurrence of is replaced by or even where . The cases and are also allowed

PROOF We prove the case involving for finite . Define



and



Then and are continuous on the interval and differentiable on the interval for every in . By the Generalized Mean-Value Theorem (...) there exists a number in such that
.

Since , if , then neccesarily , so we have



Mean Value Theorem seems a limitless tool in Analysis. Question: ? Think so. At this point, 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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Hi, the questions are: is the limit when of the same as the quotient itself? I think so because ;
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 when tends to from the right?; Wikipedia talks about an interval an tending, presumably -but not for sure- to (more intuitive to me).
 
  • #4
mcastillo356 said:
Hi, the questions are: is the limit when of the same as the quotient itself?
Only if the quotient is well defined at . It might be another limit of the form , for example.
 
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  • #5
PeroK said:
Only if the quotient is well defined at . It might be another limit of the form , for example.
Or .
 
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  • #6
PeroK said:
Only if the quotient is well defined at . It might be another limit of the form , 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:
If we differentiate top and bottom, we get another indeterminate form:
Your question, if I understand it, is that why don't we drop the limit on the RHS and evaluate the function at ? The answer is because we can't as we have another expression of the .

But, we can apply L'Hopital's rule a second time to get:
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 , if , then neccesarily , so we have



Mean Value Theorem seems a limitless tool in Analysis. Question: ? Think so. At this point, doesn´t add worth information; it's a useless limit
But is not a constant: it is a function of as it is the result of applying the mean value theorem to a function on . So it is more accurate, and perhaps clearer, to write At this point we can replace with a dummy variable (either introducing a new symbol or repurposing or ) to get
 
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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 ? The answer is because we can't as we have another expression of the .
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 . Basically they can differ in many basic limit operations.
 
  • #10
pasmith said:
But is not a constant: it is a function of as it is the result of applying the mean value theorem to a function on
Brilliant, thanks indeed for the post, I really thought I was facing reals. Nice remark
 

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