What is the logic behind L'Hopital's Rule?

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Discussion Overview

The discussion revolves around the rationale and underlying logic of L'Hopital's Rule, focusing on its theoretical justification and proofs rather than its application. Participants explore different perspectives on understanding the rule, including its proof and conceptual foundations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that L'Hopital's Rule has a solid proof, suggesting that engaging with the proof can enhance understanding of limits.
  • Others describe L'Hopital's Rule as a shortcut, likening it to other derivative and integral formulas, and mention that it can be proven using epsilon-delta definitions.
  • A participant proposes using Taylor series approximations to explain L'Hopital's Rule, detailing how the limit can be approached through the derivatives of the functions involved.

Areas of Agreement / Disagreement

Participants express differing views on the nature of L'Hopital's Rule, with some emphasizing its proof and others considering it a shortcut. No consensus is reached regarding a singular rationale behind the rule.

Contextual Notes

Some discussions involve assumptions about the behavior of functions near specific points, such as the requirement that both functions approach zero. The exploration of Taylor series as a method to understand the rule introduces additional mathematical considerations that remain unresolved.

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Can anyone tell me, what is the rationale behind L'Hopital's Rule? I just know that how to use it but don't know why it is logic.
 
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its not any logic ... it has good enough proof ...
try proving it if u want ... shouldn't be hard ...
it should give u a good rundown on all the properties of limits :)

-- AI
 
CartoonKid said:
Can anyone tell me, what is the rationale behind L'Hopital's Rule? I just know that how to use it but don't know why it is logic.

It's possible to prove the various versions of L'Hopital's rule using epsilons and deltas. It's really, more or less, a shortcut like all of the derivative and integral formulae.
 
One easy way to think of L'Hopital's rule in its most usual form, is to consider the Taylor series approximations of the two functions:
Suppose a function [tex]f(x)\approx{f}(a)+f'(a)(x-a)[/tex]
in a neighbourhood of x=a. (This is a first-order Taylor-approx.)
Similarly, we have a function [tex]g(x)\approx{g}(a)+g'(a)(x-a)[/tex]
Hence we have that [tex]h(x)=\frac{f(x)}{g(x)}\approx\frac{f(a)+f'(a)(x-a)}{g(a)+g'(a)(x-a)}[/tex]
in the same neighbourhood.
We are interested in [tex]\lim_{x\to{a}}h(x)[/tex]
Furthermore, we assume f(a)=g(a)=0, that is:
[tex]h(x)\approx\frac{f'(a)}{g'(a)}[/tex] close enough.
L'Hopital's rule states that this is, in fact the limit of h(x) as x goes to a.
 
Last edited:
Thank you all of you for your contribution. Finally, I got it.
 

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