limit of absolute value $$ \lim_{x \to 0} \frac{\lvert2x-1\rvert - \lvert2x+1\rvert}{x} $$ Defining the function piecewise reveals the limit is in fact, continuous about 0 However when I go to solve it in a normal algebraic manner, the $2x$ terms are canceling, leaving me with an undefined output - 0 in the denominator. Any hints would be fantastic, I've solved this every way I can think of and I keep getting different answers, none of which are the correct answer. I did graph this, and it does show a continuous function around $0$ where $f(x) = -4$ My problem here is that I'm being a huge dunce about absolute values.

For $\;x\;$ pretty close to zero, $\;2x-1<0\;,\;\;2x+1>0\;$ , so we have the limit

$$\frac{-2x+1-2x-1}x=\frac{-4x}x=-4\xrightarrow[x\to 0]{}-4$$