Science:Math Exam Resources/Courses/MATH110/April 2019/Question 01 (c)
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Question 01 (c) 

Let Is continuous at ? Explain why. 
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 

Given a function , and a point in the domain of , what does it mean for the function to be continuous at the point ? This will involve checking left and righthand limits, but should include more information than this. 
Hint 2 

In order to determine whether the function given in the question is continuous at , you will need to compare the following quantities:
itself, no limits here. In order for to be continuous, what must be true about these three quantities, and how must they be related to one another? 
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 are asked to determine whether the function defined in the problem statement is continuous at the point We'll see that the function is discontinuous at . The easiest way to see this is to compare the left and righthand limits of the function at the point . A simple calculation gives: and . Therefore, the left and righthand limits are not equal. Examining the definition of continuity, we see that must be discontinuous at , since these onesided limits are not equal. 