Science:Math Exam Resources/Courses/MATH100/December 2015/Question 07
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Question 07 

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! 
Hint 

Recall the definition of differentiability at some point. 
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. 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
