Answered

Find solutions to your problems with the expert advice available on IDNLearn.com. Ask your questions and receive prompt, detailed answers from our experienced and knowledgeable community members.

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 = []).

Use Definitions Of Righthand And Lefthand Limits To Prove The Limit Statementlim1xXgt0Since X Approaches 0 From The Left Xlt0 X class=

Sagot :

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

[tex]|x|=\begin{cases}x,x\ge0 \\ \\ -x,x<0\end{cases}[/tex]

|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...

[tex]\lim _{x\rightarrow0-}\frac{x}{|x|}=\lim _{x\rightarrow0-}\frac{x}{-x}=\lim _{x\rightarrow0-}(-1)=-1[/tex]

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

[tex]\lim _{x\rightarrow0-}\frac{x}{|x|}=\lim _{x\rightarrow0-}\frac{x}{-x}[/tex]

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

View image DelaineyL449171
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.