is the discontinuity at x = - 3 in the given graph removable? explain. choose the correct answer below. a…

is the discontinuity at x = - 3 in the given graph removable? explain. choose the correct answer below. a. no, because the function approaches the same value to the left and right of x = - 3. b. yes, because f(-3)=lim f(x) as x→ - 3. c. no, because f cannot be made continuous at x = - 3 by redefining f(-3). d. yes, because the function is defined for x = - 3.

is the discontinuity at x = - 3 in the given graph removable? explain. choose the correct answer below. a. no, because the function approaches the same value to the left and right of x = - 3. b. yes, because f(-3)=lim f(x) as x→ - 3. c. no, because f cannot be made continuous at x = - 3 by redefining f(-3). d. yes, because the function is defined for x = - 3.

Answer

Explanation:

Step1: Recall definition of removable discontinuity

A removable discontinuity at (x = a) exists when (\lim_{x\rightarrow a}f(x)) exists (left - hand limit equals right - hand limit), but (f(a)) is either not defined or not equal to the limit. If we can re - define (f(a)) to be equal to (\lim_{x\rightarrow a}f(x)), the function becomes continuous at (x = a).

Step2: Analyze given options

Option A: If the function approaches the same value from the left and right of (x=-3), it is a candidate for removable discontinuity, not a non - removable one, so A is incorrect. Option B: If (f(-3)=\lim_{x\rightarrow - 3}f(x)), the function is continuous at (x = - 3), not having a removable discontinuity. Option C: A removable discontinuity can be made continuous by redefining the function value at the point of discontinuity. If it cannot be made continuous by redefining (f(-3)), it is non - removable, which is correct. Option D: Just because the function is defined at (x=-3) does not mean it has a removable discontinuity.

Answer:

C. No, because f cannot be made continuous at (x = - 3) by redefining (f(-3)).