Problem 14 - 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). $$ \begin{array}{l} f(x)=\frac{1}{x-2} ; \lim _{x \rightarrow 2} f(x) \\ \hline \boldsymbol{x} \quad 1.9 \quad 1.99 \quad 1.999 \quad 2.001 \quad 2.01 \quad 2.1 \\ \hline \boldsymbol{f}(\boldsymbol{x}) \end{array} $$

Step-by-Step Solution

Answer

\(f(x) = \frac{1}{x-2}\) for \(x = 1.9, 1.99, 1.999, 2.001, 2.01, 2.1\): \(f(1.9) \approx -10, \: f(1.99) \approx -100, \: f(1.999) \approx -1000, \: f(2.001) \approx 1000, \: f(2.01) \approx 100, \: f(2.1) \approx 10\). As x approaches 2, f(x) becomes increasingly larger (in both the positive and negative directions) and does not approach a specific value. Therefore, the limit does not exist.

Step 1: Compute f(x) for given values of x

Let's calculate f(x) for each value of x provided in the table \( (1.9, 1.99, 1.999, 2.001, 2.01, 2.1) \). After plugging in each x value into the function, we will compute the corresponding f(x).

Step 2: Analyse the table values

Analyse the computed f(x) values when x approaches 2 from both the left and the right, i.e., using the values \(1.999, 1.99, 1.9 \) and the values \(2.001, 2.01, 2.1\).

Step 3: Estimate the limit

Examine the behavior of f(x) as x approaches 2 and use this analysis to estimate the value of the limit, if it exists.

Key Concepts

Function Evaluation

To begin understanding limits, we first need to evaluate the function at specific points. Function evaluation is the process of substituting a value for the variable in a function and calculating the result. For our exercise, we are given the function \(f(x) = \frac{1}{x-2}\) and need to find \(f(x)\) for the given values of \(x\).

• 1.9 -> \(f(1.9) = \frac{1}{1.9 - 2} = -10\)
• 1.99 -> \(f(1.99) = \frac{1}{1.99 - 2} = -100\)
• 1.999 -> \(f(1.999) = \frac{1}{1.999 - 2} = -1000\)
• 2.001 -> \(f(2.001) = \frac{1}{2.001 - 2} = 1000\)
• 2.01 -> \(f(2.01) = \frac{1}{2.01 - 2} = 100\)
• 2.1 -> \(f(2.1) = \frac{1}{2.1 - 2} = 10\)

These calculations show us how the function behaves as \(x\) gets closer to 2.

Estimate Limits

After evaluating the function at given points, the next step is to estimate the limit. Limits help us understand how a function behaves as the input approaches a particular value.
In our case, the aim is to estimate \(\lim_{x \to 2} f(x)\).
Looking at our function values:

• As \(x\) approaches 2 from the left (1.9, 1.99, 1.999), \(f(x)\) becomes increasingly negative: -10, -100, -1000.
• As \(x\) approaches 2 from the right (2.001, 2.01, 2.1), \(f(x)\) becomes increasingly positive: 1000, 100, 10.

This behavior suggests a vertical asymptote at \(x = 2\), indicating that the limit does not exist at this point since the function approaches infinity from one side and negative infinity from the other.

Mathematical Analysis

Mathematical analysis allows us to look deeper into functions and their limits. By analyzing the behavior of the function \(f(x) = \frac{1}{x-2}\), we predict the function's trend as \(x\) approaches 2.
A key concept is continuity: a function is continuous at a point if the limit at that point exists and equals the function's value. Our function, however, is not continuous at \(x = 2\) because \(f(x)\) is undefined at this point.
Since the function values diverge towards infinity and negative infinity depending on the direction of approach, there is no single value it converges to. Thus, \(\lim_{x \to 2} f(x)\) does not exist. This is a classic example of a function having a vertical asymptote, a line that the graph approaches but never touches or crosses. Understanding this behavior is fundamental in calculus, emphasizing the need to analyze functions from both left and right approaches.