Question

Evaluate the limit of x/(absolute value of x) as x approaches 0- (from the left side).
???

Answer 1

\[\Large \left(|x|\right)' \quad=\quad \frac{x}{|x|}\]

Answer 2

\[
f(x) = \frac x {|x|}
\]

Answer 3

Since \(x\leq 0\) we know \(|x| = -x\)\[
\frac x{|x|}\to \frac x{-x} \to -1
\]

Answer 4

But if x\[\le0\], shouldn't that -x be \[\left| -x \right|\]?

Answer 5

Suppose \(x=-1<0\)
Then \(|x|=|-1| = 1 = -x\)

Answer 6

I see what you're saying now.

Answer 7

The thing is that \(-x\) doesn't make \(x\) negative.
It changes the sign of \(x\).
Negative \(x\) values need to change sign to become positive.