Answer
$\lim\limits_{\Delta x\to 0}\dfrac{\cos{(\pi+\Delta x)}+1}{\Delta x}=0.$
Work Step by Step
$\lim\limits_{\Delta x\to 0}\dfrac{\cos{(\pi+\Delta x)}+1}{\Delta x}$
$=\lim\limits_{\Delta x\to 0}\dfrac{\cos{\pi}\cos{\Delta x}-\sin{\pi}\sin{\Delta x}+1}{\Delta x}$
$=\lim\limits_{\Delta x\to 0}\dfrac{1-\cos{\Delta x}}{\Delta x}=0.$
