Question

Removable and Nonremovable Discontinuities Describe the difference between a discontinuity that is removable and one that is nonremovable. In your explanation, give examples of the following descriptions. (a) A function with a nonremovable discontinuity at \(x=4\) (b) A function with a removable discontinuity at \(x=-4\) (c) A function that has both of the characteristics described in parts (a) and (b)

Short answer

A nonremovable discontinuity is a point where the function does not exist and it can't be fixed, for example \( f(x) = 1/(x-4) \) at \( x = 4 \). A removable discontinuity is a point where the function does not exist but can be fixed by redefining a single point, for example \( g(x) = (x^2+16)/(x+4) \) at \( x = -4 \). A function with both discontinuities example is \( h(x) = f(x) + g(x) = 1/(x-4) + (x^2+16)/(x+4) \).

Step-by-step solution

Nonremovable discontinuity example

A nonremovable discontinuity can be created by defining a function with a domain that inherently excludes certain x-values. For example \( f(x) = 1/(x-4) \) has a nonremovable discontinuity at \( x = 4 \). Because as \( x \) approaches 4, the function goes to infinity, creating a vertical asymptote, which cannot be removed or fixed.

Removable discontinuity example

A removable discontinuity occurs when a certain x-value is undefined in the function, but the limit of the function as x approaches that value is finite. Consider \( g(x) = (x^2+16)/(x+4) \). For \( x = -4 \), the function is undefined, creating a hole. But the limit as \( x \) approaches -4 is \(4\), so this discontinuity is removable.

Function with both discontinuities

To demonstrate a function with both types of discontinuities, simply add the two above functions together. So, \( h(x) = f(x) + g(x) = 1/(x-4) + (x^2+16)/(x+4) \). This function still has a nonremovable discontinuity at \( x = 4 \) and a removable discontinuity at \( x = -4 \)

Key concepts

Removable Discontinuity

Imagine you're walking along a path and you come across a small gap that you can easily step over. In mathematics, this is akin to a removable discontinuity in a function. A removable discontinuity occurs at a certain point on the graph where the function is undefined, however, the limits from both sides approach the same value. In other words, if you could hypothetically 'fill in' this single point, the function would be continuous.

For illustration, consider the function g(x) = (x^2+16)/(x+4). When x = -4, the denominator becomes zero and the function is undefined, creating a 'hole' in the graph. Despite this, as x approaches -4 from either side, the limit remains 4. This suggests that if we were to define g(-4) = 4, we could 'remove' the discontinuity, thus it is considered removable.

Nonremovable Discontinuity

Contrast a removable discontinuity with a situation where our path abruptly stops at the edge of a cliff. A similar concept exists in math, known as a nonremovable discontinuity. This type occurs when the function’s limit does not exist at a certain point, or when the two one-sided limits exist but are not equal to each other. This interruption in the function can't be simply 'fixed' by filling in a single point.

A classic example is the function f(x) = 1/(x-4). As x approaches 4, the function values grow without bound, indicating that the function has a vertical asymptote at x = 4. This behavior creates a nonremovable discontinuity since you cannot define a finite point to make the function continuous at x = 4. Thus, a nonremovable discontinuity reflects an inherent, unbridgeable gap in the function's graph.

Limit of a Function

Understanding limits is crucial when dealing with discontinuities. The limit of a function at a certain point is the value that the function's output approaches as its input approaches that point. You can think of it as what the y-value should be as x gets incredibly close to a certain number from both the left and right sides.

A limit can be finite like in the case of removable discontinuities, or it may be infinite, which often occurs at nonremovable discontinuities such as vertical asymptotes. Calculating limits helps us understand the behavior of functions at points that are not explicitly defined or at points approaching infinity, providing insight into the function's overall continuity and potential discontinuities.

Vertical Asymptote

When studying limits and discontinuities, you'll occasionally encounter functions with vertical asymptotes. These are like invisible barriers that the function approaches but never touches or crosses. Mathematically, a vertical asymptote is a line where the function's limit approaches positive or negative infinity as the input approaches a certain value.

For instance, with the function f(x) = 1/(x-4), as x comes closer to 4, the function's values increase or decrease without limit, indicating a vertical asymptote at x = 4. Vertical asymptotes are associated with nonremovable discontinuities because, at these points, the function does not reach a certain finite value but instead becomes infinitely large in magnitude.