Suppose 𝑔(𝑥)={𝑥2+2𝑥+1𝑥+1𝑖𝑓 𝑥<12𝑥𝑖𝑓 𝑥≥1. g(x)={ x+1x 2 +2x+1 2x if x<1if x≥1 . The best description concerning the continuity of g(x) is that the function:A.has a removable discontinuity.B.has an infinite discontinuity.C.has both infinite and removable discontinuities.D.has a jump discontinuity.E.is continuous.
Question
Suppose 𝑔(𝑥)={𝑥2+2𝑥+1𝑥+1𝑖𝑓 𝑥<12𝑥𝑖𝑓 𝑥≥1. g(x)={ x+1x 2 +2x+1 2x if x<1if x≥1 . The best description concerning the continuity of g(x) is that the function:A.has a removable discontinuity.B.has an infinite discontinuity.C.has both infinite and removable discontinuities.D.has a jump discontinuity.E.is continuous.
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 form, which is x=1.
First, let's find the limit of g(x) as x approaches 1 from the left (x<1). This corresponds to the first part of the function definition, which is x^2 + 2x + 1.
lim (x->1-) g(x) = lim (x->1-) (x^2 + 2x + 1) = 1^2 + 2*1 + 1 = 4
Next, let's find the limit of g(x) as x approaches 1 from the right (x>=1). This corresponds to the second part of the function definition, which is 2x.
lim (x->1+) g(x) = lim (x->1+) (2x) = 2*1 = 2
Since the two one-sided limits are not equal, the function g(x) is not continuous at x=1. This is a jump discontinuity because the function jumps from one value to another at x=1. Therefore, the best description concerning the continuity of g(x) is that the function has a jump discontinuity. So, the correct answer is D. has a jump discontinuity.
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