Science:Math Exam Resources/Courses/MATH110/April 2016/Question 03 (a)
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Question 03 (a)
Consider the following function, where is a constant.
For , which one of the following statements is the best reason why is not continuous at ?
(i) does not exist.
(ii) does not exist.
(iii) Both (i) and (ii).
(iv) and exist, but they are not equal.
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Recall that a function is continuous at a point if
1, the function is defined at the point
2 the limit of the function exists at the point
3.the limit is equal to the value of the function at the point.
If one of the conditions fails, we say that the function is discontinuous at the point.
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First, we note that the function is defined at with , so (i) and (iii) are not correct.
Now let’s see whether the limit of the function at exists. For this purpose, we consider the left limit and the right limit. When approaches to from the right, we consider points near which is greater than . For such points, the corresponding piece of the function is , so that
Here we use the given information .
On the other hand, to get the left limit, we consider points close to . Since the corresponding piece of the function in such region is , we get
Since the left limit is not equal to right limit at ,
the limit does not exist. In all, we choose .