Question

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.

Explanation:

Step1: Recall definition of removable discontinuity

A removable discontinuity at \(x = a\) exists when \(\lim_{x
ightarrow a}f(x)\) 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 \(\lim_{x
ightarrow a}f(x)\), 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 \(f(-3)=\lim_{x
ightarrow - 3}f(x)\), 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 \(f(-3)\).