Question

Limit Question:\[
\lim_{x\to0}\frac{x}{|x|}
\]My hypothesis is that it doesn't exist.

Answer 1

Problem is that l'hospital doesn't get me anywhere. \[
\frac{x'}{|x|'} = \frac{1}{\frac{x}{|x|}} = \frac{x}{|x|}
\]

Answer 2

One thing I have noticed is that: \[
\frac{x}{|x|} = \begin{cases}
1&\quad&x>0\\
-1&&x<0
\end{cases}
\]

Answer 3

So this means it can't exist because left and right hand don't exist.

Answer 4

I mean the left and right hands exist, but don't converge

Answer 5

Is \(\ |x|\) even a one-to-one function? If thats even a valid question to ask....

Answer 6

Well \(|x|\) is not one to one because \(|-1|=|1| = 1\), but I don't think that has any relevance when it comes to limits.

Answer 7

I mean \(|x|\to 0\) as \(x\to0\), so it doesn't matter.

Answer 8

Thats right,
|x|/x or x/|x| doesn't exist as
left and right hand limits are unequal.

Answer 9

i mean their limit doesn't exist :P

Answer 10

simply it does not exist

Answer 11

thus not differentiable at 0 :\

Answer 12

I've got another question related to limits

Answer 13

post