why does lim (x/|x|) = 1
x-> 0+
It does because we are only concerned about the behavior of the graph from the right, IE: (positive values of x)
Note how any positive number divided by the absolute value of that number will always be positve 1.
Since $\displaystyle |x|=\left\{\begin{array}{cc}{-x},&\mbox{ if }{x}\leq0\\x,&\mbox{ if }{x>0}\end{array}\right.$
Then for values greater than 0, we have $\displaystyle \frac{x}{|x|}=\frac{x}{x}$
Therefore $\displaystyle \lim_{x\to0^+}\frac{x}{|x|}=\lim_{x\to0^+}\frac{x} {x}=\lim_{x\to0^+}(1)=1$.