Problem 62 - Contemporary Precalculus

Question

Involve the greatest integer function \(f(x)=[| x] |\) which was defined in Example 7 on page \(145 .\) You may use your calculator as an aid in analyzing these problems. Let \(k(x)=\frac{x}{[| x]|+[|-x]|} ;\) find \(\lim _{x \rightarrow 1} k(x),\) if this limit exists.

Step-by-Step Solution

Answer

Question: Determine the limit of the function \(k(x) = \frac{x}{[| x]| +[|-x]|}\) as \(x\) approaches \(1\). Answer: The limit does not exist.

Step 1: Understand the function k(x)

The given function \(k(x) = \frac{x}{[| x]| +[|-x]|}\). As we can see, both the numerator and the denominator have absolute values intertwined with greatest integer functions.

Step 2: Find the value of k(x) for x > 1

For \(x > 1\), we know that \(|x|=x\) and \(|-x|= -x\). So the function \(k(x)\) becomes: \(k(x) = \frac{x}{[x] + [-x]}\) Now, considering that \(x\) is greater than \(1\), the greatest integer value of \([x]\) will be \(x\) itself, and the greatest integer value of \([-x]\) will be \(-x\). Therefore, \(k(x) = \frac{x}{x - x}= \frac{x}{0}\) which is undefined.

Step 3: Find the value of k(x) for x < 1

For \(x < 1\), we have \(|x|=-x\) and \(|-x|= x\). So the function \(k(x)\) becomes: \(k(x) = \frac{x}{[-x] + [x]}\) Since \(-1 < x < 1\), we know that the greatest integer value of \([-x]\) is \(0\) and the greatest integer value of \([x]\) is also \(0\). So we get: \(k(x) = \frac{x}{0+0} = \frac{x}{0}\), which is again undefined.

Step 4: Evaluate the limit

As we can see from steps 2 and 3, the function \(k(x)\) is undefined for both \(x>1\) and \(x<1\). Thus, the limit does not exist. Hence, the limit for the given function as \(x\) approaches \(1\) does not exist: \(\lim_{x \rightarrow 1} k(x) =\) limit does not exist.

Key Concepts

Greatest Integer Function

The greatest integer function is fundamental in discrete mathematics and analysis. It's often represented as \([| x|]\) or sometimes \( ext{floor}(x)\). This function takes any given real number and rounds it down to the nearest integer. For instance, \([| 3.9|] = 3\) and \([| -1.2|] = -2\).

• Key Points: The function rounds numbers down, regardless of their decimal part.
• Applications: Used in algorithms and when you need to work with whole numbers in calculations.

Understanding how the greatest integer function works in different intervals is essential, especially when combined with other operations like absolute values, as seen in the problem statement. In essence, for any number \(x\), \([| x |]\) returns the largest integer smaller than or equal to \(x\). This is straightforward, but it can complicate expressions when combined with limits and other nonlinear functions.

Limits

The limit of a function as it approaches a specific value is a fundamental concept in calculus. It describes the behavior of the function near that point, rather than at the point itself. In this particular problem, we're interested in \(\lim_{x \rightarrow 1} k(x)\), looking for how the function \(k(x)\) behaves as \(x\) gets close to 1 from either side.

• Understanding Limits: Limits require careful examination of the function from both sides of the target value, which can reveal discontinuities or indeterminate forms.
• Approaching 1: Here, the definitions become significant as \(x > 1\) or \(x < 1\). As \(x\) approaches 1, the function \(k(x)\) becomes undefined due to division by zero.

When functions are defined piecewise, like those involving the greatest integer function, limits help us understand the overall behavior even when \(x\) isn't exactly equal to the target value. Often, graphical or numerical methods complement analytical solutions to give a clear picture.

Undefined Expressions

In mathematics, an expression being undefined often signals an indeterminate form, such as division by zero. In the exercise regarding \(k(x)\), as \(x\) approaches 1 from either direction, both the numerator and denominator evaluate to expressions involving zero (the denominator becomes zero), which makes the function undefined at \(x = 1\).

• Key Insight: When expressions lead to division by zero, they become undefined because any real number divided by zero is not a valid computation.
• Problem-Solving: When encountering undefined expressions, double-check assumptions, limits, and whether simplification or different approaches to solve exist.

Undefined expressions are a red flag when calculating limits because they can often mean the limit does not exist. This is exactly what occurs here, where \(k(x)\) is not defined around \(x=1\), thus resulting in the limit not existing. Understanding why expressions become undefined is crucial for a well-rounded comprehension of calculus and related subjects.