Question

Use definitions of right-hand and left-hand limits to prove the limit statement.lim-1|x|X>0Since x approaches 0 from the left, x<0, (x = []).

Answer 1

First we need to understand what |x| means or what values it repressents

`|x|=\begin{cases}x,x\ge0 \\ \\ -x,x<0\end{cases}`

|x| indicates the absolute value of x, this is, x is always going to be positive, for example,

when x = 1 -> |x| = 1 , but also when x = -1 , then |x| = 1

Since, in this case, we need to find the limit when X approaches 0 from the left we are going to use |x| = -x , for x<0

this is...

`\lim _(x\rightarrow0-)(x)/(|x|)=\lim _(x\rightarrow0-)(x)/(-x)=\lim _(x\rightarrow0-)(-1)=-1`

At this point we have proved the limit statement.

So, in order to answer the question in the lower part... x approaches to 0 from the left, x<0, |x| = -x

In the graph you can see, whenever X<0 the value of the funcion will be negative and when it approaches 0 it becomes -1

On the other hand, when the function approaches to 0 from the right, the value of the function is +1. This is a discontinuity

`\lim _(x\rightarrow0-)(x)/(|x|)=\lim _(x\rightarrow0-)(x)/(-x)`

This way we eliminate the absolute value, because, remember, when x<0, |x| = -x