how to find $\lim_{x\to0}{\frac{|2x-1|-|2x+1|}{x}}=-4$ $$\lim_{x\to0}{\frac{|2x-1|-|2x+1|}{x}}=-4$$
Why? This came from a calculus book, before L'hopital is introduced. I couldn't find the answer myself, so I looked at the answers page. WolframAlpha agrees, and interestingly enough, the function is equal to $-4$ in the entire range $[-0.5,0.5]$, so maybe you could use squeeze theorem (which has been introduced) to evaluate the limit? Here is some of my working so far.
$$\lim_{x\to0}{\frac{|2x-1|-|2x+1|}{x}}\\=2\lim_{x\to0}{\frac{|x-0.5|-|x+0.5|}{x}}\\=2\lim_{y\to0.5}{\frac{|y-1|-|y|}{y-0.5}}$$
The last step substitutes with $y=x+0.5$. It is the step at which I am stuck. Squeeze theorem? Thanks.

Hint: Try to see the limits in two cases where once $x \to 0^+$ and other $x \to 0^-$.
