- #1

Leo Liu

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- Homework Statement:
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- Relevant Equations:
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Problem: Prove that $$\lim_{x\to 0}|\frac{|x|} x=\text{Undefined}$$

The solution written by my prof. uses a special case where the tolerance of error ##\epsilon=1/2##. However, I want to proof it with generality, meaning that the tolerance is shrinking.

Below is my attempt at a solution:

$$0<|x-0|<\delta;\, |f(x)-L|<\epsilon,\,\forall \epsilon>0$$

$$0<x^2<\delta^2;\, (f(0)-L)^2<\epsilon^2$$

$$\begin{cases}

f(x>0)=1\\

f(x<0)=-1

\end{cases}$$

This is where I am stuck. It looks the two part of the function are independent of ##\delta##, making it impossible to establish a relationship with ##\epsilon##. Can somebody please help?

The solution written by my prof. uses a special case where the tolerance of error ##\epsilon=1/2##. However, I want to proof it with generality, meaning that the tolerance is shrinking.

Below is my attempt at a solution:

$$0<|x-0|<\delta;\, |f(x)-L|<\epsilon,\,\forall \epsilon>0$$

$$0<x^2<\delta^2;\, (f(0)-L)^2<\epsilon^2$$

$$\begin{cases}

f(x>0)=1\\

f(x<0)=-1

\end{cases}$$

This is where I am stuck. It looks the two part of the function are independent of ##\delta##, making it impossible to establish a relationship with ##\epsilon##. Can somebody please help?

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