Science:Math Exam Resources/Courses/MATH102/December 2014/Question A 02
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Question A 02 

What value of makes the function below continuous? Hint: Writing down the definition of the derivative of at might be useful here. 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Recall that a function is continuous at if What is the we must consider here, and what is the limit of as 
Hint 2 

We observe that is continuous for all so for to be continuous (everywhere), it suffices to have What are and 
Hint 3 

What is To evaluate this limit, you may wish to recall that the derivative of a function is by definition Alternatively, you could try using the squeeze theorem, l'Hôpital's rule, or Taylor series. 
Checking a solution serves two purposes: helping you if, after having used all the hints, 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. We observe that is continuous for all (as it is the quotient of continuous functions), so for to be continuous (everywhere), it suffices to have Now while Hence for continuity we must have This limit can be evaluated in numerous ways; we choose to follow the hint in the question statement and observe that by definition since But we know that so the value of the limit is Therefore we take 