MATH110 April 2016
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • Q4 (c) • Q4 (d) • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q5 (e) • Q5 (f) • Q5 (g) • Q5 (h) • Q5 (i) • Q6 (a) • Q6 (b) • Q7 • Q8 (a) • Q8 (b) • Q8 (c) • Q8 (d) • Q9 (a) • Q9 (b) • Q10 •
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 ?
(iii) Both (i) and (ii).
(iv) and exist, and they are equal.
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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.
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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 .