Suppose𝑔(𝑥)={1𝑥−2𝑖𝑓𝑥<12𝑥−4𝑖𝑓𝑥≥1.g(x)={ x−21 2x−4 ifx<1ifx≥1 .The best description concerning the continuity of g(x) is that the function:A.is continuous.B.has a jump discontinuity.C.has an infinite discontinuity.D.has a removable discontinuity.E.None of the above
Question
Suppose𝑔(𝑥)={1𝑥−2𝑖𝑓𝑥<12𝑥−4𝑖𝑓𝑥≥1.g(x)={ x−21 2x−4 ifx<1ifx≥1 .The best description concerning the continuity of g(x) is that the function:A.is continuous.B.has a jump discontinuity.C.has an infinite discontinuity.D.has a removable discontinuity.E.None of the above
Solution
To determine the continuity of the function g(x), we need to examine the behavior of the function at the point where it changes definition, which is x = 1.
First, we calculate the limit of g(x) as x approaches 1 from the left (x < 1) using the first definition of g(x):
lim (x->1-) g(x) = 1*(1) - 2 = -1
Next, we calculate the limit of g(x) as x approaches 1 from the right (x >= 1) using the second definition of g(x):
lim (x->1+) g(x) = 2*(1) - 4 = -2
Since the two one-sided limits are not equal, the function g(x) has a jump discontinuity at x = 1. Therefore, the correct answer is B. "has a jump discontinuity".
Similar Questions
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