Science:Math Exam Resources/Courses/MATH110/April 2016/Question 03 (b)
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Question 03 (b) 

Consider the following function, where is a constant. For , which one of the following statements is the best reason why is continuous at ? (i) exists. (ii) exists. (iii) Both (i) and (ii). (iv) and exist, and they are equal. 
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 the continuity of a function at a point (See the Hint of the part (a) in this question.) Note that all three conditions should be satisfied to be continuous. 
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 note that the function is defined at in the third piece of piecewise function, and When , let’s see the left limit and right limit at point as follows: and We get that the left limit is equal to right limit, thus exists. And note that which is the reason that is continuous at . So we choose .
