Problem 16 - 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). $$ \begin{array}{l} f(x)=\frac{x-1}{x-1} ; \lim _{x \rightarrow 1} f(x) \\ \hline \boldsymbol{x} \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) & \multicolumn{3}{|c} {} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The completed table is: $$ \begin{array}{l} f(x)=\frac{x-1}{x-1} ; \lim _{x \rightarrow 1} f(x) \\\ \hline x \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\\ \hline f(x) \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \\\ \hline \end{array} $$ Based on the table, we can see that as x approaches 1, \(f(x)\) remains constant at 1, and thus the limit as \(x \rightarrow 1\) of the function \(f(x) = \frac{x-1}{x-1}\) is \(1\).

Step 1: Calculate \(f(x)\) at given x values

To fill in the table, we must evaluate the function \(f(x)\) at each of the given x values: 0.9, 0.99, 0.999, 1.001, 1.010, and 1.1. Remember that the function is \(f(x) = \frac{x-1}{x-1}\). Notice that if \(x \neq 1\), we can simplify the function as follows: \(f(x) = \frac{x-1}{x-1} = 1\) Now, let's calculate \(f(x)\) at the given values of x: \(f(0.9) = 1\) \(f(0.99) = 1\) \(f(0.999) = 1\) \(f(1.001) = 1\) \(f(1.01) = 1\) \(f(1.1) = 1\)

Step 2: Complete the table

With our calculated values of \(f(x)\), the table now looks like this: $$ \begin{array}{l} f(x)=\frac{x-1}{x-1} ; \lim _{x \rightarrow 1} f(x) \\\ \hline \boldsymbol{x} \quad 0.9 \quad 0.99 \quad 0.999 \quad 1.001 \quad 1.01 \quad 1.1 \\\ \hline \boldsymbol{f(x)} \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \quad 1 \\\ \hline \end{array} $$

Step 3: Determine the limit as x approaches 1

From the table, we can see that as x approaches 1, \(f(x)\) remains constant at 1. This suggests that the limit as x approaches 1 exists, and the limit is: \(\lim_{x \rightarrow 1}f(x) = 1\) Thus, the limit as x approaches 1 of the function \(f(x) = \frac{x-1}{x-1}\) is 1.

Key Concepts

Applied Mathematics

At its core, applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry. Thus, the development of computational tools like calculating the limit of a function plays a crucial role in practical problem solving.

In the given exercise, applied mathematics comes into play when we consider the function evaluation at numbers close to 1, which can be thought of as a simulation or model to predict the behavior of the function at that point. By approximating the function's value as x approaches 1, we use a fundamental tool of applied mathematics to predict and understand the behavior of systems at critical points.

Calculating Limits

The process of calculating limits involves approaching a particular point from both sides of the x-axis and observing the behavior of the function values. It is a fundamental concept in calculus and analysis, serving as the groundwork for defining continuity, derivatives, and integrals.

In the given problem, as all the evaluated points when substituted into the function give a result of 1, we intuitively predict the limiting value through pattern recognition. Because each output was consistent irrespective of how we approached x = 1, our estimation of the limit is made with confidence. Calculating limits is about understanding how a function behaves as the input gets arbitrarily close to a particular value, not just at that value.

Continuous Functions

A function is said to be continuous at a point if the limit of the function as x approaches that point equals the function's value at that point. Intuitively, this means that the graph of a continuous function can be drawn without lifting the pen from the paper.

For the limit to exist, the function values must approach a single number from both sides of the point in question. In our exercise, we see that the limit as x tends to 1 is 1, which coincides with what the function value would be at x=1 if we define it correctly, signaling that the function is continuous at that point (barring the fact that the original function is not defined at x=1).

Function Evaluation

The concept of function evaluation is straightforward: plug a number into the function, and get a result out. When the input number gets closer to a specific value, function evaluation provides insight into the function's behavior near that point, which is exactly what we did in our exercise.

The calculated results from the function were used to fill out the provided table and illustrate the behavior of the function as x gets closer to 1. Through repeated function evaluations, we can establish a pattern or trend which aids in predicting the value of the limit. It's crucial not only to evaluate a function at various points but also to understand what those evaluations mean in the broader context of limit and continuity.