Science:Math Exam Resources/Courses/MATH110/April 2018/Question 02 (a)
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Question 02 (a) 

Determine for what values of is continuous. 
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 

Remember that when and are continuous at , so is if . 
Hint 2 

Check that the denominator is nonzero and that the value inside the square root is nonnegative. 
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. Remember that the functions and are each continuous everywhere. First, we check the denominator is continuous on the real line. Note that a composite function of two continuous functions is also continuous on its domain. Remember that is continuous on its domain . On the other hand, since is always at least , will always be at least zero. i.e., it is in the domain of the square root function. Therefore, the denominator (which is a composite of and ) is welldefined in the real line, so it is continuous on the real line. Now, the only remaining thing to check is that the denominator is nonzero. is nonzero if and only if , and if and only if is an odd integer multiple of . Since the numerator is nonzero for these values of , we conclude that is continuous for all except , where is an integer. Answer: is continuous at any . 