I need to find the values of $a$ at which $\lim_{h \to 0} \frac{\sin(a+h)-\sin(a)}{h} = 0.$ I know that this means that we are looking for the values of $a$ at which $\dfrac{d}{dx}\sin x=0$, or $\cos x=0$. I also know from calculus 1 that a should equal $\dots,\dfrac{\pi}{2}, \dfrac{3\pi}{2}, \dots$
However, I can't figure out how I can show this using "real-analysis" level definitions and theorems. Do I need to use the definition of limits to show that $\frac{\sin(a+h)-\sin(a)}{h} \to 0$? How do I use that to find the appropriate values of $a$?