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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 $a$?

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    Ahhh yes. I do remember that. I guess I could use that proof to show that the derivative of sinx is cosx and then show the values of a at which cosx=0? Thank you so much for your guidance!2012-11-11

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