Trying To Use Squeeze Theorem To Prove Derivatives Are Equal

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Homework Help Overview

The discussion revolves around the application of the Squeeze Theorem to demonstrate that the derivatives of three functions are equal at a specific point. The original poster attempts to establish this by showing that the limits of the functions converge to the same value as they approach that point.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the use of the Squeeze Theorem and the limit definition of derivatives. There are questions about the validity of the original poster's reasoning and whether their approach sufficiently proves the desired result.

Discussion Status

Some participants express skepticism about the original poster's conclusions, suggesting that the argument presented does not constitute a proof. There is mention of an alternative approach involving an epsilon-delta argument, which may provide a more rigorous framework for the discussion.

Contextual Notes

Participants note that the problem statement is referenced but not provided in detail, which may limit the clarity of the discussion. There is also a suggestion to utilize LaTeX for mathematical expressions to enhance communication.

toslowtogofast2a
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Homework Statement
Problem Statement In Image Below
Relevant Equations
Squeeze Theorem and Limit Definition of Derivative
Below is the question and my attempt at a solution. From the info in the problem I tried to use the squeeze thm to show limf(x)=limg(x)=limh(x) all as x goes to a. Then that combined with f(a)=g(a)=h(a) I used to say all 3 derivatives are equal. Is my attempt below correct or did I make an error somewhere along the way. Thanks in advance.
SQUEEZE 2A.JPG
 
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Without losing generality we can do
a=0, f(a)=g(a)=h(a)=0.
When x>0
\frac{f(x)}{x} \leq \frac{g(x)}{x} \leq \frac{h(x)}{x}
x ##\rightarrow## +0
f'(0) \leq \frac{g(x)}{x} \leq h'(0)
Do similar when x <0
 
Last edited:
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toslowtogofast2a said:
Homework Statement: Problem Statement In Image Below
Relevant Equations: Squeeze Theorem and Limit Definition of Derivative

Below is the question and my attempt at a solution. From the info in the problem I tried to use the squeeze thm to show limf(x)=limg(x)=limh(x) all as x goes to a. Then that combined with f(a)=g(a)=h(a) I used to say all 3 derivatives are equal. Is my attempt below correct or did I make an error somewhere along the way. Thanks in advance.
View attachment 355021
First of all, allow me to invite you to use the LaTeX capabilities of Physics Forums when posting mathematical expressions. Find a link to the LaTeX Guide here or at the end of this Thread. You are likely to get better responses to your posts if you use LaTeX.
 
toslowtogofast2a said:
Homework Statement: Problem Statement In Image Below
Relevant Equations: Squeeze Theorem and Limit Definition of Derivative

Below is the question and my attempt at a solution. From the info in the problem I tried to use the squeeze thm to show limf(x)=limg(x)=limh(x) all as x goes to a. Then that combined with f(a)=g(a)=h(a) I used to say all 3 derivatives are equal. Is my attempt below correct or did I make an error somewhere along the way. Thanks in advance.
View attachment 355021
I would say you haven't proved anything. You've more or less just stated that the result is obvious.
 
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PS the question hints at using a direct ##\epsilon-\delta## approach, which might be a good exercise.
 

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