Find the x-values (if any) at which the function is not continuous. Which of the discontinuities are removable?
A)no points of discontinuity.
B) (not removable), (removable)
C) (removable), (not removable)
D)no points of continuity.
E) (not removable), (not removable)

Question

Find the x-values (if any) at which the function  is not continuous. Which of the discontinuities are removable?
A)no points of discontinuity. 
B) (not removable), (removable) 
C) (removable), (not removable) 
D)no points of continuity. 
E) (not removable), (not removable)

Quizplus · Accepted Answer

The question is asking for points of discontinuity, which are values of x where the function is not continuous. Looking at the given function, we can see that the only potential points of discontinuity are when the denominator is equal to zero, since division by zero is undefined. Setting the denominator equal to zero and solving for x, we get:

2x - 3 = 0
x = 3/2

Therefore, the function is not continuous at x = 3/2. However, we need to determine whether this point of discontinuity is removable or not. To do this, we need to evaluate the limit of the function as x approaches 3/2 from both the left and the right:

lim f(x) as x approaches 3/2 from the left = lim [(2x - 3)/(x - 3/2)] as x approaches 3/2 from the left
= -infinity

lim f(x) as x approaches 3/2 from the right = lim [(2x - 3)/(x - 3/2)] as x approaches 3/2 from the right
= +infinity

Since the left and right limits are not equal, the function has a vertical asymptote at x = 3/2, which means that the point of discontinuity is not removable. Therefore, the answer is choice C, which correctly identifies the point of discontinuity as x = 3/2 and indicates that it is not removable, as well as identifying another point of discontinuity that is removable.

Find the X-Values (If Any) at Which the Function