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.

\begin{align}
\lim_{h\to 0}\frac{\sin(\pi+h)}{h(\pi+h)}&=-\lim_{h\to 0}\frac{\sin h}{h} \cdot\frac1{\pi+h}\\
&=-\lim_{h\to 0}\frac1{\pi+h}\\
&=-\frac1\pi
\end{align}
