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Below is an AP calculus question. Then answer key says the answer is E. But I don't understand if the function is differentiable and f '(3) = 5, then how could the limit not also be 5? I think that if choice E was false then the function is not differentiable.

Does any know of a differentiable function where f '(3) = 5, but the limit of f '(x) at 3 is not also 5?

Or is there something wrong with this question?

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Michael Maliszesky
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    For a function to be differentiable at a point it needs to be continuous at that point. There is however no requirement that the function needs to be twice differentiable at a point, and no requirement that the derivative be continuous at a point. – JMoravitz Sep 28 '22 at 14:44
  • Consider modifying the example found here by appropriate translations. – JMoravitz Sep 28 '22 at 14:50

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$$f(x)=\left\{\begin{array}{cc}(x-3)^2 \sin\left( \frac1{x-3}\right ) + 5x-7&&,&x \neq 3\\8&&,&x=3\end{array}\right\}$$

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Ben G.
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