Find the x-values (if any)at which the function is not continuous.Which of the discontinuities are removable?

Question

Find the x-values (if any)at which the function  is not continuous.Which of the discontinuities are removable? ​

Quizplus · Accepted Answer

To identify the points of discontinuity, we need to check where the function is undefined or where the limit does not exist. Looking at the function, we notice that there is a variable in the denominator. For a fraction, the denominator cannot be zero. Therefore, the function is undefined at any value of x that makes the denominator zero.

Setting the denominator equal to zero, we get:
(x-3)(x+2)(x-1) = 0 
which has roots at x=3, x=-2, and x=1.

Therefore, the function is not continuous at x=3, x=-2, and x=1.

To determine if any of these discontinuities are removable, we need to check if the limits exist at these points. Evaluating the limits of the function as x approaches each of these points, we find:

lim(x-->3) 11eac44f_ede8_0de9_bedf_a1418ccc8776_TB7497_11 = 5/18
lim(x-->-2) 11eac44f_ede8_0dea_bedf_5fd85dbf31df_TB7497_11 = DNE
lim(x-->1) 11eac44f_ede8_0deb_bedf_6275d6d93e66_TB7497_11 = -5/2

The limit at x=3 exists and is finite, so the discontinuity at x=3 is removable. The limit at x=1 exists but is not equal to the function value, so the discontinuity at x=1 is also removable. However, the limit at x=-2 does not exist or is infinite, so the discontinuity at x=-2 is not removable.

Therefore, the function has removable discontinuities at x=3 and x=1, and a non-removable discontinuity at x=-2, giving us choice C as the correct answer.

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