Trying To Use Squeeze Theorem To Prove Derivatives Are Equal

[toslowtogofast2a]

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.

 

[anuttarasammyak]

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

 

[SammyS]

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.

 

[PeroK]

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.

 

[PeroK]

PS the question hints at using a direct $\epsilon-\delta$ approach, which might be a good exercise.