Problem 43 - Calculus Early Transcendentals

Question

Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, find a function \(g\) that agrees with \(f\) for \(x \neq a\) and is continuous at a. (a) \(f(x)=\frac{x^{4}-1}{x-1}, \quad a=1\) (b) \(f(x)=\frac{x^{3}-x^{2}-2 x}{x-2}, \quad a=2\) (c) \(f(x)=[\sin x], \quad a=\pi\)

Step-by-Step Solution

Answer

Functions (a) and (b) have removable discontinuities; function (c) does not.

Step 1: Identify the Removable Discontinuity

A removable discontinuity occurs when a function has a hole at a point, which can be filled by redefining the function at that point. To identify this, look for places where the function's formula can be simplified at the given point.

Step 2: Analyze Function (a)

Given: \( f(x) = \frac{x^4 - 1}{x - 1} \) at \( a = 1 \).1. Factor the numerator: \(x^4 - 1 = (x^2 + 1)(x^2 - 1) = (x^2 + 1)(x - 1)(x + 1)\).2. Simplify: \( f(x) = \frac{(x^2 + 1)(x - 1)(x + 1)}{x - 1} = (x^2 + 1)(x + 1) \) for \( x eq 1 \).3. Define \( g(x) = (x^2 + 1)(x + 1) \).4. \( g(x) \) is continuous at \( x = 1 \).Function (a) has a removable discontinuity at \( a = 1 \) and \( g(x) = (x^2 + 1)(x + 1) \) is continuous at \( x = 1 \).

Step 3: Analyze Function (b)

Given: \( f(x) = \frac{x^3 - x^2 - 2x}{x - 2} \) at \( a = 2 \).1. Factor the numerator: \(x^3 - x^2 - 2x = x(x^2 - x - 2) = x(x - 2)(x + 1)\).2. Simplify: \( f(x) = \frac{x(x - 2)(x + 1)}{x - 2} = x(x + 1) \) for \( x eq 2 \).3. Define \( g(x) = x(x + 1) \).4. \( g(x) \) is continuous at \( x = 2 \).Function (b) has a removable discontinuity at \( a = 2 \) and \( g(x) = x(x + 1) \) is continuous at \( x = 2 \).

Step 4: Analyze Function (c)

Given: \( f(x) = [\sin x] \) at \( a = \pi \), where \([.\)] denotes the greatest integer function.1. Note that \( [\sin x] \) rounds the sine value down to the nearest integer.2. \( \sin \pi = 0 \); small changes around \( \pi \) can lead \([\sin x]\) to fluctuate between 0 and -1.Function (c) has a jump discontinuity at \( a = \pi \). It is not removable because the function jumps between discrete values depending on whether \( \sin x \) is slightly positive or slightly negative.

Key Concepts

Greatest Integer Function

The greatest integer function, also known as the floor function, is a piecewise function that maps any real number to the largest integer not greater than the number itself. You write it as \( \[x\] \), where \(x\) is a real number.

• This function poses unique challenges in continuity because it's inherently discontinuous wherever the input is not an integer.
• For example, \(\sin x\) is a smooth and continuous function. However, when wrapped in the greatest integer function like \([\sin x]\), it will "jump" to the nearest integer, causing disruptions at certain points.

This creates a jump discontinuity rather than a removable one. When examining the function \(\sin x\) at \(a = \pi\), \(\sin \pi = 0\), but small fluctuations around \(\pi\) swing between 0 and -1, making the discontinuity non-removable.

Factoring Polynomials

Factoring polynomials is a crucial skill in algebra, used for simplifying expressions and solving equations. When you factor a polynomial, you express it as a product of simpler polynomials. This becomes particularly useful in identifying removable discontinuities in rational functions.

• Consider the function \(f(x) = \frac{x^4 - 1}{x - 1}\). Its numerator can be factored as \((x^2 + 1)(x^2 - 1) = (x^2 + 1)(x - 1)(x + 1)\), which allows the cancellation of the \(x - 1\) term.
• Similarly, for \(f(x) = \frac{x^3 - x^2 - 2x}{x - 2}\), factoring yields \(x(x - 2)(x + 1)\), enabling the cancellation of \(x - 2\).

These cancellations indicate a removable discontinuity at the points where the function was initially undefined due to division by zero. The essential takeaway is that factoring can simplify functions, revealing hidden continuity.

Continuous Functions

Continuous functions are ones that have no breaks, jumps, or holes in their graph. For a function \(g(x)\) to be continuous at a point \(a\), three conditions must be met:

• The function \(g(x)\) is defined at \(x = a\).
• The limit of \(g(x)\) as \(x\) approaches \(a\) from both directions exists.
• The limit of \(g(x)\) as \(x\) approaches \(a\) is equal to \(g(a)\).

When addressing removable discontinuities, the trick is to redefine the function at critical points to fill in gaps. For example, redefining by using limits can convert seemingly discontinuous functions such as \(f(x) = \frac{x^4 - 1}{x - 1}\) and obtaining \(g(x) = (x^2 + 1)(x + 1)\) that is continuous at \(x = 1\). This provides a solution by bridging the continuity.

Sine Function Properties

The sine function is a periodic and continuous function, known for its smooth, wave-like pattern which repeats every \(2\pi\). It is defined for all real numbers and has several key properties that make it unique:

• The range of the sine function is \([-1, 1]\), meaning it never exceeds these bounds.
• At integer multiples of \(\pi\), the sine function crosses the x-axis, resulting in values of 0.

However, when the greatest integer function is applied to \(\sin x\), the smoothness of the sine wave is disrupted. Particularly at \(x = \pi\), where \[\sin x\]\ is concerned, the integer function causes a jump discontinuity, shifting the values abruptly between integers. This alteration transforms the normally uninterrupted sine curve into a step-like graph whenever approached by the greatest integer function.