## Describe the difference between a discontinuity that is removable and one that is nonremovable. In your explanation, give specific examples of the following: a. A function with nonremovable discontinuity at x = 3. b. A function with a removable discontinuity at x = -3. c. A function that has both a removable and a nonremovable discontinuity.

Jan 30, 2018

See explanation.

#### Explanation:

A discontinuity at $x = c$ is removable if ${\lim}_{x \to c} f \left(x\right)$ exists but ${\lim}_{x \to c} f \left(x\right) \ne f \left(c\right)$.

A discontinuity at $x = c$ is nonremovable if ${\lim}_{x \to c} f \left(x\right)$ does not exist for any of a variety of reasons. The limits from the left or right or both could be positive or negative infinity. The limit from the left and right could both be finite but not be equal.

a) $f \left(x\right) = \frac{1}{x - 3}$

b) $f \left(x\right) = \frac{x + 3}{x + 3}$

c) $f \left(x\right) = \frac{x - 3}{\left(x - 3\right) \left(x - 4\right)}$