Question

Removable and Nonremovable Discontinuities In Exercises \(35-60,\) find the \(x\) -values (if any) at which \(f\) is not continuous. Which of the discontinuities are removable? \ $$ f(x)=\frac{|x+7|}{x+7} $$

Short answer

The function \( f(x)=\frac{|x+7|}{x+7} \) is not continuous at \( x=-7 \). The discontinuity at \( x=-7 \) is not removable; it's a nonremovable discontinuity.

Step-by-step solution

Find the x value where the function is undefined

Firstly, identify the x value that makes the denominator zero. Set the denominator equal to zero and solve for x. For the function \( f(x)=\frac{|x+7|}{x+7} \), this gives \( x+7=0 \). Solving this equation gives \( x=-7 \).

Check the continuity of the function at x=-7

Check if the function is continuous at x=-7. The function is discontinuous at x=-7 since the denominator equals zero. It creates a hole in the function at this point.

Determine the type of discontinuity at x=-7

It remains to determine if the discontinuity at x=-7 is removable or not. For this, compute the limit of the function as x approaches -7 from both directions. If both limits exist and are equal, the discontinuity is removable. Otherwise, it is nonremovable. Given the absolute function in the numerator of the given function, the limit from the left will result in -1 while the limit from the right will result in 1. Since the two limits are not equal, the discontinuity at x=-7 is not removable; it's a nonremovable discontinuity.

Key concepts

Removable Discontinuity

In calculus, a removable discontinuity occurs when a function is not continuous at a point, yet the limit exists at that point because the function approaches the same value from the left and the right. Visually, this might look like a tiny hole in the graph of the function where the function value at the discontinuous point can be redefined to make the function continuous.

For example, consider the function given in the exercise \( f(x)=\frac{|x+7|}{x+7} \). When \( x=-7 \), the function is not defined, creating a discontinuity. However, if the limits as \( x \) approaches -7 from both sides were equal, we could 'fill in the hole' by redefining the function value at \( x=-7 \). This would convert a removable discontinuity into a point of continuity, by simply redefining \( f(-7) \) to be the value that the limits approach. Unfortunately, in this case, the limits from either side of -7 are not equal, indicating that this discontinuity cannot be remedied by redefining the function at \( x=-7 \).

Nonremovable Discontinuity

Contrasting with removable discontinuities, a nonremovable discontinuity indicates a more severe 'break' in the function where the limit does not exist, or exists but differs when approached from the left and right. This can manifest as a jump, an infinite discontinuity, or an endpoint where the function is not defined and cannot be made continuous by simply redefining the function at a point.

In the textbook exercise, \( f(x)=\frac{|x+7|}{x+7} \), we find a nonremovable discontinuity at \( x=-7 \). As we approach \( x=-7 \) from the left, the function approaches -1, while from the right, it approaches 1. Since the two one-sided limits are not equal, the function has a jump discontinuity at \( x=-7 \) that cannot be 'fixed' to make the function continuous there.

Continuity of Functions

Discussing continuity of functions involves understanding where a function is uninterrupted or unbroken over its domain. A function is continuous at a point if the function's value at that point is equal to the limit of the function as it approaches that point. In other words, the graph of a continuous function can be drawn without lifting the pencil from the paper.

Continuity checks generally involve three conditions: the function must be defined at the point, the limit must exist at that point, and the function's value must equal that limit. The previous analysis of \( f(x)=\frac{|x+7|}{x+7} \) at \( x=-7 \) shows that while \( f(x) \) is defined for all \( x \eq -7 \) and the limit exists from both directions, the lack of equality between the function's value and the limits at \( x=-7 \) breaks continuity there.

Limits in Calculus

In calculus, limits help us study the behavior of functions as they approach specific points or infinity. Limits in calculus are foundational to the concept of continuity and are key to understanding and identifying discontinuities. A limit can be thought of as the value that a function 'approaches' as its input (or \( x \) value) gets arbitrarily close to some number.

Computing limits, as in the exercise with \( f(x)=\frac{|x+7|}{x+7} \), often reveals the nature of discontinuities. The limit of \( f(x) \) as \( x \) approaches -7 does not exist because the behavior of the function differs depending on which direction \( x \) is approaching -7 from. Therefore, identifying limits can aid in classifying whether a discontinuity is removable or nonremovable and in understanding the overall behavior of a function around points of interest.