Problem 57 - 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 \(h(x)=[x]+[-x]\); find \(\lim _{x \rightarrow 2} h(x),\) if this limit exists.

Step-by-Step Solution

Answer

Answer: The limit of the function \(h(x) = [x] + [-x]\) as \(x\) approaches \(2\) is \(0\).

Step 1: Understand the Function h(x)

First, let's understand the function \(h(x) = [x] + [-x]\). This function is a combination of the greatest integer functions for \(x\) and \(-x\). That means for any \(x\), the function will output the sum of the greatest integers for both \(x\) and \(-x\).

Step 2: Identify the Input Value for the Limit

Our task is to find the limit of the function \(h(x)\) as \(x\) approaches \(2\). So, we are interested in finding the value, if it exists, of \(\lim_{x\rightarrow 2} h(x)\).

Step 3: Break Down the Function Into Parts

To calculate the limit, we need to break down the function \(h(x)\) into its parts and find the greatest integer values for \(x\) and \(-x\), respectively. For the given value of \(x=2\), \[ [2] = 2\] \[[-2] = -2\] Now, we can find the sum of the greatest integers for both \(x\) and \(-x\): \[ h(2) = [2] + [-2] = 2 + (-2) = 0\]

Step 4: Determine the Limit

As \(x\) approaches \(2\), \(h(x)\) approaches the value \(0\). Therefore, the limit of the function \(h(x)\) as \(x\) approaches \(2\) exists and is equal to \(0\): \[\lim_{x\rightarrow 2} h(x) = 0\] The limit exists, and its value is \(\boxed{0}\).

Key Concepts

Greatest integer function

The greatest integer function, often denoted as \([x]\), is a special type of function known as the floor function. It rounds down a real number to the closest integer less than or equal to that number. For example:

• \([5.7] = 5\)
• \([-2.3] = -3\)

These examples show that the function effectively floors the number to the nearest integer that is less or equal to it. This concept becomes important when dealing with expressions that involve both \(x\) and its negative counterpart \(-x\), like in our function \(h(x) = [x] + [-x]\). Understanding how each part contributes to the sum aids in grasping how the function behaves as \(x\) changes.

Piecewise functions

A piecewise function is defined by different expressions based on different parts of its input domain. It usually looks like a series of conditional expressions. Although \(h(x) = [x] + [-x]\) doesn't appear as a piecewise function at first glance, its behavior can change at different points due to the nature of the floor function.

Behavior Analysis with Piecewise Thinking

To fully understand \(h(x)\), consider how it will act for various values of \(x\):

• If \(x\) is an integer, \([x] = x\) and \([-x] = -x\), so \(h(x) = 0\).
• If \(x\) is not an integer, \([x] = x - \text{small fraction}\) and \([-x] = -x - \text{small fraction}\), resulting in \(h(x) = -1\).

Recognizing this behavior helps in understanding how the function might behave as \(x\) approaches certain values.

Calculating limits at a point

Limits at a point try to determine what value a function approaches as its input nears a particular point. It's a fundamental concept in calculus that tells us about a function’s behavior right at or around that point.

Steps to Calculate Limits

For the function \(h(x) = [x] + [-x]\) approaching \(x = 2\):

• For \(x = 2\), \([2] = 2\) and \([-2] = -2\), making \(h(2) = 0\).
• Consider values of \(x\) near, but not exactly \(2\), such as just below and above 2 to see if the outputs consistent with zero.

The consistency when these values are analyzed signifies that the function smoothly approaches the value \(0\). Therefore, the limit \(\lim_{x \rightarrow 2} h(x)\) indeed exists and equals 0. To sum up, these steps help clarify the limit existence and value at precise points of interest.