1.10 - Exploring Types of Discontinuities

The graph of the function f is shown above. What are all values of x for which f has a removable discontinuity?

0 and 2 only A removable discontinuity occurs at x=c if limx→cf(x)=L exists but f(c) is not defined or f(c)≠L. For this function, limx→0f(x) exists but f(0) is not defined, and limx→2f(x) exists but f(2) is not defined. Therefore, there are removable discontinuities at x=0 and at x=2.

The graph of the function f is shown above. What are all values of x for which f has a removable discontinuity?

1 and 7 only A removable discontinuity occurs at x=c if limx→cf(x)=L exists but f(c) is not defined or f(c)≠L. For this function, limx→1f(x) exists but f(1) is not defined, and limx→7f(x) exists but f(7)≠limx→7f(x)=0. Therefore, there are removable discontinuities at x=1 and at x=7.

Let f be the function defined by f(x)=x3−2x2−3xx3−3x2+4. Which of the following statements about f at x=2 and x=−1 is true?

f has a discontinuity due to a vertical asymptote at x=2 , and f has a removable discontinuity at x=−1 . A graphing calculator is used to graph the function f. The graph suggests that there is a discontinuity due to a vertical asymptote at x=2. To verify this, let g(x)=x3−2x2−3x and h(x)=x3−3x2+4 so that f(x)=g(x)h(x). There is a discontinuity due to a vertical asymptote at x=2 because g(2)=−6≠0 and h(2)=0. The graph might appear to be continuous at x=−1. However, there is a removable discontinuity at x=−1. limx→−1f(x) does exist, but f(−1) is not defined because g(−1)=h(−1)=0.

Let f be the function defined by f(x)=x3−9xx3+x2−8x−12. Which of the following statements about f at x=−2 and x=3 is true?

f has a discontinuity due to a vertical asymptote at x=−2 , and f has a removable discontinuity at x=3 . A graphing calculator is used to graph the function f. The graph suggests that there is a discontinuity due to a vertical asymptote at x=−2. To verify this, let g(x)=x3−9x and h(x)=x3+x2−8x−12 so that f(x)=g(x)h(x). There is a discontinuity due to a vertical asymptote at x=−2 because g(−2)=10≠0 and h(−2)=0. The graph might appear to be continuous at x=3. However, there is a removable discontinuity at x=3. limx→3f(x) does exist, but f(3) is not defined because g(3)=h(3)=0.

Let f be the function defined by f(x)=3x3+2x2x2−x. Which of the following statements is true?

f has a removable discontinuity at x=0 and a discontinuity due to a vertical asymptote at x=1 . The function f is not defined at x=0 because the denominator equals 0 when x=0. However, limx→0f(x) exists, as shown below. Therefore, f has a removable discontinuity at x=0. limx→0f(x)=limx→03x3+2x2x2−x=limx→0x(3x2+2x)x(x−1)=limx→03x2+2xx−1=3⋅02+2⋅00−1=0 The graph of the rational function f has a vertical asymptote at x=1 because the numerator is nonzero and the denominator equals 0 when x=1.

Let f be the function defined by f(x)=x4−4x2x2−4x. Which of the following statements is true?

f has a removable discontinuity at x=0 and a discontinuity due to a vertical asymptote at x=4 . The function f is not defined at x=0 because the denominator equals 0 when x=0. However, limx→0f(x) exists, as shown below. Therefore, f has a removable discontinuity at x=0. limx→0f(x)=limx→0x4−4x2x2−4x=limx→0x2(x2−4)x(x−4)=limx→0x(x+2)(x−2)x−4=0(0+2)(0−2)(0−4)=0 The graph of the rational function f has a vertical asymptote at x=4 because the numerator is nonzero and the denominator equals 0 when x=4.