$x/|x|$ question about division

Question

What is $\frac{x}{|x|}$ can it be simplified? Because look at this.

$\frac{r\cosh(x)}{\sqrt{\cosh^2(x)}} = \frac{r\cosh(x)}{|\cosh(x)|}$

How do you do this?

Answer 1

When $x>0$, $\frac{x}{|x|}=\frac{x}{x}=1$.

When x=0, it is indeterminate due to division by 0.

When $x<0$, $\frac{x}{|x|}=\frac{x}{-x}=-1$

Note that cosh(x) is always greater than 0 so you can remove the absolute value, thus, $\frac{r \cosh x}{|\cosh x|}=\frac{r \cosh x}{\cosh x}=r$

Answer 2

If $x>0$ then $|x|=x$ so that $\frac{x}{|x|}=1$.

If $x<0$ then $|x|=-x$ so that $\frac{x}{|x|}=-1$.

If $x=0$ then $|x|=0$ so that $\frac{x}{|x|}$ is not defined.