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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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It might help us if you told us your definition of sine. – EuYu Nov 11 '12 at 4:52
My definition of sin? I'm not sure if I understand the request. – Jess Nov 11 '12 at 5:26
How are you defining the sine function? That will impact how the proof is given. – EuYu Nov 11 '12 at 5:31
I'm not sure if this answers your question at all, but sinx is defined for all real numbers. – Jess Nov 11 '12 at 5:37
You're interested in a "real analysis" level proof. That means you have to start with with a rigorous definition of the sine function. So again I ask, what is the sine function? Are you defining it as a power series? In terms of the complex exponential? As the solution of a differential equation? – EuYu Nov 11 '12 at 5:42

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