MATH100 December 2015
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Is the function
differentiable at ? You must explain your answer using the definition of the derivative.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Recall the definition of differentiability at some point.
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First, we check that is continuous at 0. (Note that if is not continuous at 0, is not differentiable at 0.)
Using , we can compute the right hand limit as
On other hand, it is easy to get
Therefore, is continuous at 0.
Now, we show is differentiable at 0.
By the definition of derivative, first consider the right hand limit:
The last inequality follows from
Then, the left hand limit can be computed as
Since the right and left hand limits coincide, the limit exists so that is differentiable at 0 and .