Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Complete the table by computing \(f(x)\) at the given values of \(x\). Use these results to estimate the indicated limit (if it exists). $$ \begin{array}{l} f(x)=\frac{|x|}{x} ; \lim _{x \rightarrow 0} f(x)\\\ \begin{array}{lllllll} \hline \boldsymbol{x} & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & & & & & & \\ \hline \end{array} \end{array} $$

Step-by-Step Solution

Answer

The computed values of the function \(f(x)=\frac{|x|}{x}\) at the given \(x\) values are: $$ \begin{array}{l} f(x)=\frac{|x|}{x} ; \lim _{x \rightarrow 0} f(x)\\\ \begin{array}{lllllll} \hline \boldsymbol{x} & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\\ \hline \boldsymbol{f}(\boldsymbol{x}) & -1 & -1 & -1 & 1 & 1 & 1 \\\ \hline \end{array} \end{array} $$ As the function has different limits when approaching 0 from the left and the right (\(\lim_{x\rightarrow 0^-}f(x)=-1\) and \(\lim_{x\rightarrow 0^+}f(x)=1\)), we conclude that the limit does not exist as \(x\) approaches 0.

Step 1: Compute f(x) for each table value

We need to calculate \(f(x)\) for the given values of \(x\) in the table. To do this, we will substitute each value of \(x\) into the function and simplify. For \(x = -0.1\), \(f(x) = \frac{|-0.1|}{-0.1} = \frac{0.1}{-0.1} = -1\). For \(x = -0.01\), \(f(x) = \frac{|-0.01|}{-0.01} = \frac{0.01}{-0.01} = -1\). For \(x = -0.001\), \(f(x) = \frac{|-0.001|}{-0.001} = \frac{0.001}{-0.001} = -1\). For \(x = 0.001\), \(f(x) = \frac{|0.001|}{0.001} = \frac{0.001}{0.001} = 1\). For \(x = 0.01\), \(f(x) = \frac{|0.01|}{0.01} = \frac{0.01}{0.01} = 1\). For \(x = 0.1\), \(f(x) = \frac{|0.1|}{0.1} = \frac{0.1}{0.1} = 1\). Now, we can fill in the table as follows: $$ \begin{array}{l} f(x)=\frac{|x|}{x} ; \lim _{x \rightarrow 0} f(x)\\\ \begin{array}{lllllll} \hline \boldsymbol{x} & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\\ \hline \boldsymbol{f}(\boldsymbol{x}) & -1 & -1 & -1 & 1 & 1 & 1 \\\ \hline \end{array} \end{array} $$

Step 2: Estimate the limit as x approaches 0

To estimate the limit of \(f(x)\) as \(x\) approaches 0, we will observe the behavior of the function values as \(x\) gets closer to 0 from both the left and the right sides. Looking at the table, we can see that when \(x\) approaches 0 from the left (negative values), the function values are all -1. When \(x\) approaches 0 from the right (positive values), the function values are all 1. Thus, we have: \(\lim_{x\rightarrow 0^-}f(x)=-1\) \(\lim_{x\rightarrow 0^+}f(x)=1\) Since the function has different limits when approaching 0 from the left and the right, we conclude that the limit does not exist as \(x\) approaches 0.

Key Concepts

Understanding Absolute Value Functions

An absolute value function is a mathematical function that returns the non-negative value of a number irrespective of its sign. This function is commonly denoted as \( |x| \), which reads as 'the absolute value of x'.

In the context of our exercise, we used the absolute value function in a fraction with \( x \), to find \( f(x) = \frac{|x|}{x} \). Calculating this function can be very insightful, especially as \( x \)--approaches zero.

Understanding how \( |x| \)--functions is crucial when evaluating limits, particularly when \( x \)--takes on values that are very close to 0, as it allows us to see how the function behaves when dealing with both positive and negative values of \( x \).

Let us take a practical approach to grasp the concept. For any positive value of \( x \), the absolute value function won't change its value, which means \( |x| = x \). However, for any negative value of \( x \), the absolute value turns it into a positive one, because \( |-x| = x \). Hence, when \( x \)--is negative, \( \frac{|x|}{x} \) simplifies to -1, and when \( x \)--is positive, the simplification results in 1, demonstrating a clear distinction in the function's output depending on the sign of \( x \).

Deciphering One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a certain point from only one side - either from the left or the right. These are crucial in understanding the behavior of functions around points of interest and are notated as \( \lim_{x\to c^-}f(x) \) for the left-hand limit, and \( \lim_{x\to c^+}f(x) \) for the right-hand limit, where \( c \) represents the point being approached.

In our exercise scenario, the one-sided limits were calculated as \( x \) approaches 0. From the left-side limit (as \( x \) approaches 0 from negative values), we found that the limit was -1, notated as \( \lim_{x\rightarrow 0^-}f(x)=-1 \). Conversely, from the right-side limit (as \( x \) approaches 0 from positive values), the limit was 1, written as \( \lim_{x\rightarrow 0^+}f(x)=1 \). This discrepancy is essential in determining if the limit at a point exists, as a limit only exists at a point if and only if the left-hand and right-hand limits are equal.

Limits and Continuity at a Glance

The concepts of limits and continuity are interconnected, forming the bedrock of calculus. A limit is the value that a function approaches as the input gets closer to some number. Continuity, on the other hand, is a property of a function if it is continuous at every point in its domain; that is, the graph of the function is unbroken.

For a function to be continuous at a point \( c \), the limit of the function as it approaches \( c \) must equal the function's value at \( c \) itself; symbolically, \( \lim_{x\to c}f(x) = f(c) \).

When dealing with the limit of \( \frac{|x|}{x} \) as \( x \) approaches 0, we note that the function \( f(x) \) is not continuous at \( x=0 \) because the one-sided limits do not match. The left and right limits are -1 and 1, respectively, so they do not converge to a single value. Therefore, we can conclude that the function has a discontinuity at \( x=0 \), reaffirming that the limit as \( x \) approaches 0 does not exist. Knowing this, it becomes apparent why a solid grasp of limits and continuity is indispensable for understanding the behavior of functions within calculus.