Answer
$\lim\limits_{\Delta x\to0}\dfrac{(x+\Delta x)^2-4(x+\Delta x)-(x^2-4x)}{\Delta x}=2x-4.$
Work Step by Step
$\lim\limits_{\Delta x\to0}\dfrac{(x+\Delta x)^2-4(x+\Delta x)-(x^2-4x)}{\Delta x}$
$=\lim\limits_{\Delta x\to0}\dfrac{x^2+2x(\Delta x)+(\Delta x)^2-4x-4\Delta x-x^2+4x}{\Delta x}$
$=\lim\limits_{\Delta x\to0}\dfrac{\Delta x(2x-4+\Delta x)}{\Delta x}$
$=\lim\limits_{\Delta x\to0}(2x-4+\Delta x)=2x-4+0=2x-4.$
