Determining a limit without L'Hospital's Rule

I'm trying to solve the limit $$\lim_{h\to0}\frac{\sin(\pi+h)}{h(\pi+h)}$$ without using L'Hospital's rule. It's part of a problem for which I am trying to prove that $$\lim_{h\to0}\frac{\frac{\sin(\pi+h)}{|\pi+h|}-\frac{h}{\pi}}{|h|}$$ exists. After a bit of simplifying and assuming $h>0$ I have arrived at the first expression.