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Each limit represents the derivative of some function $ f $ at some number $ a $. State such an $ f $ and $ a $ in each case.

$ \displaystyle \lim_{\theta \to \pi/6} \frac{\sin \theta - \frac{1}{2}}{\theta - \pi/6} $

$f(\theta)=\sin \theta$

$a=\frac{\pi}{6}$

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Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Let's talk about this question. We are given a limit and this limit actually represents the derivative of some function f. At some number eight. Uh So if you remember the derivative is nothing but the slope. So if this is the co and this is one of the point A comma F. And this is another point B comma F B. Then it's derivative is nothing. But the slope which is F B minus F or b minus A. But if a approaches to be a approaches to be or in other words, a N B overlap. So that becomes a tangent. And then we represented as a limit uh is approaching towards B is approaching towards B uh F B minus F or b minus. So that's the required derivative. So if you compare it but does limit. So we are we are given that the function is scientific because data is now approaching two priority. So we're going to say that the function there's nothing but the scientist and the value of A is nothing but uh F B minus F. So F. A. When we're gonna port which value of tita is such that when we put over here we get 1/2. So that's clearly pi over six. Because if you put by over six signs by over six is one or two or scientific degrees one or two. So the value of A. Which we are going to use this by over six. So this actually represents the derivative of the function scientific to at equal to five or six. Thank you