Problem 72 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Complete the table by computing \(f(x)\) at the given values of \(x\). Use the results to guess at the indicated limits, if they exist. $$ f(x)=\frac{|x|}{x} ; \lim _{x \rightarrow \infty} f(x) \text { and } \lim _{x \rightarrow-\infty} f(x) $$

Step-by-Step Solution

Answer

Based on our evaluation of the function \(f(x)=\frac{|x|}{x}\), we can conclude that the limit as \(x\) approaches \(\infty\) is 1, and the limit as \(x\) approaches \(-\infty\) is -1: \(\lim_{x\rightarrow\infty} f(x) = 1\) \(\lim_{x\rightarrow -\infty} f(x) = -1\)

Step 1: Understand the function\(f(x)=\frac{|x|}{x}\)

We are given the function \(f(x)=\frac{|x|}{x}\). Here, \(|x|\) represents the absolute value of \(x\). The absolute value function is defined as: \[|x|= \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}\] Now let's evaluate the function at some particular values of \(x\).

Step 2: Evaluate the function at different values of \(x\)

To gain some understanding of the function's behavior, let's evaluate it at different values of \(x\), such as 1, -1, 0. 1) For \(x = 1\): \(f(1)=\frac{|1|}{1} = \frac{1}{1} = 1\) 2) For \(x = -1\): \(f(-1)=\frac{|-1|}{-1} = \frac{1}{-1} = -1\) 3) For \(x = 0\): The function is not defined at \(x = 0\), because the denominator is 0.

Step 3: Find the limit as \(x\) approaches \(\infty\)

Now, we want to find the limit as \(x\) approaches \(\infty\). As \(x\) becomes very large, the absolute value of \(x\) also becomes very large. But since \(x > 0\) when approaching \(\infty\), the absolute value of \(x\) is equal to \(x\), so we have: \(\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty} \frac{x}{x}\) This is the same as: \(\lim_{x\rightarrow\infty} \frac{x}{x} = \lim_{x\rightarrow\infty} 1 = 1\) So the limit as \(x\) approaches \(\infty\) is 1.

Step 4: Find the limit as \(x\) approaches \(-\infty\)

Now, we want to find the limit as \(x\) approaches \(-\infty\). As \(x\) becomes very large in the negative direction, the absolute value of \(x\) becomes very large and positive. Thus, we have: \(\lim_{x\rightarrow -\infty}f(x)=\lim_{x\rightarrow -\infty} \frac{-x}{x}\) This is the same as: \(\lim_{x\rightarrow -\infty} \frac{-x}{x} = \lim_{x\rightarrow -\infty} -1 = -1\) So the limit as \(x\) approaches \(-\infty\) is -1.

Step 5: Final answer:

Based on our evaluation, we can conclude that: \(\lim_{x\rightarrow\infty} f(x) = 1\) \(\lim_{x\rightarrow -\infty} f(x) = -1\)

Key Concepts

Absolute Value Function

The absolute value function, denoted as \(|x|\), plays a crucial role in mathematics. It measures the distance of a number from zero on the number line, ignoring the direction. This function can be defined piecewise as follows:
\[|x|= \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}\]
This means that the absolute value of a positive number is the number itself, while for a negative number, it is the number with its sign reversed. When you apply absolute value in functions like \(f(x)=\frac{|x|}{x}\), it helps illustrate how the function responds differently based on whether \(x\) is positive or negative. Understanding this distinction is essential for evaluating functions involving absolute values.

Evaluating Functions

Evaluating a function means finding its output values at specific input values. For the function \(f(x)=\frac{|x|}{x}\), we can evaluate it at different \(x\) values to understand its behavior. Let's consider a few examples:

• For \(x = 1\), the function becomes \(f(1)=\frac{|1|}{1} = 1\). The output is 1 because both numerator and denominator equal \(x\).
• For \(x = -1\), the function is \(f(-1)=\frac{|-1|}{-1} = -1\). Here, the sign of \(-1\) in the denominator changes the result to -1.
• For \(x = 0\), the function is undefined because it involves division by zero.

Evaluating the function across various values helps us see the consistent patterns and outcomes, enhancing our understanding of its nature.

Infinity Limits

Infinity limits in calculus explore the behavior of functions as they approach very large positive or negative values. For the function \(f(x)=\frac{|x|}{x}\), understanding limits at infinity provides insights into its long-term behavior.

As \(x\) approaches positive infinity, the limit evaluates as:
\[\lim_{x\rightarrow\infty}f(x)=\lim_{x\rightarrow\infty} \frac{x}{x} = 1\]
Since both numerator and denominator grow equally, the limit simplifies to 1.

When \(x\) approaches negative infinity, the situation changes:
\[\lim_{x\rightarrow -\infty}f(x)=\lim_{x\rightarrow -\infty} \frac{-x}{x} = -1\]
Here, the negative sign affects the outcome, leading to a limit of -1.
Understanding these limits helps predict the global behavior of the function as \(x\) grows large in either direction.

Undefined Function Behavior

Sometimes functions include expressions that can't be evaluated for specific values; these cases include division by zero. For \(f(x)=\frac{|x|}{x}\), when \(x = 0\), the denominator becomes zero, making the function undefined at this point. This undefined behavior is crucial because:

• It highlights discontinuities in the function's graph.
• Serves as a reminder of the limitations when working with mathematical expressions.
• Encourages careful consideration of the domain where a function is valid and applicable.

Understanding where and why a function becomes undefined is fundamental in calculus, helping avoid errors and deepening comprehension of mathematical concepts.