Problem 12 - Calculus Early Transcendentals

Question

$$ \begin{aligned} \text { (a) Use a graph of } & \\ & f(x)=\left(1-\frac{2}{x}\right)^{x} \end{aligned} $$ $$ \begin{array}{l}{\text { to estimate the value of } \lim _{x \rightarrow \infty} f(x) \text { correct to two }} \\ {\text { decimal places. }} \\\ {\text { (b) Use a table of values of } f(x) \text { to estimate the limit to }} \\ {\text { four decimal places. }}\end{array} $$

Step-by-Step Solution

Answer

(a) The limit is approximately 0.14. (b) The limit is approximately 0.1353.

Step 1: Understand the Problem

We need to estimate the limit of the function \( f(x) = \left(1 - \frac{2}{x}\right)^x \) as \( x \to \infty \). In part (a), we'll use a graph for a two-decimal estimate, and in part (b), we'll calculate specific values for a four-decimal estimate.

Step 2: Analyzing the Function

Let's analyze \( f(x) = \left(1 - \frac{2}{x}\right)^x \). As \( x \to \infty \), the expression \( \frac{2}{x} \to 0 \). This makes \( 1 - \frac{2}{x} \to 1 \), raising an indeterminate form \( 1^\infty \). Such forms often relate to the natural exponential function \( e \).

Step 3: Construct and Interpret the Graph

Graph \( f(x) \) using either a graphing calculator or software like Desmos. Observe as \( x \to \infty \), looking for the value \( f(x) \) approaches. Check near large \( x \) values, such as 50, 100, 200, etc., to refine the estimation.

Step 4: Estimate Limit Using the Graph

From the graph, as \( x \) grows larger, \( f(x) \) stabilizes around a constant value. Visually estimate this horizontal asymptote to two decimal places. You might see values approaching approximately 0.14.

Step 5: Create a Table of Values

Calculate specific \( f(x) \) values for large \( x \), e.g., \( x = 50, 100, 150, 200 \). Use these values to identify the trend as \( x \to \infty \). Ensure calculations are accurate to four decimal places.

Step 6: Estimate Limit Using the Table

Observe the table. If \( f(50) = 0.1383 \), \( f(100) = 0.1353 \), \( f(150) = 0.1351 \), and \( f(200) = 0.1350 \), note how these values converge. The limit estimate to four decimal places is about 0.1353.

Key Concepts

Graphical Estimation

When estimating limits graphically in calculus, it's all about using the visual aspects of a function's graph to predict the behavior as the independent variable heads towards a specific value, like infinity. With our function, \( f(x) = \left(1 - \frac{2}{x}\right)^x \), we need to observe how \( f(x) \) behaves as \( x \to \infty \). By plotting this function using a graphing tool, we look for a horizontal line where the graph appears to settle. This line is often called a horizontal asymptote.

To estimate more accurately:

• Zoom in on the section where the graph flattens out.
• Observe the y-values as \( x \) increases, focusing on larger x-values like 50, 100, or further.

By doing so, you will often be able to predict that the graph approaches a constant value, helping estimate limits to a couple of decimal places. For this function, \( f(x) \) appears to stabilize around 0.14.

Indeterminate Forms

In calculus, indeterminate forms are expressions where direct substitution during limit evaluation leads to an ambiguous form. One classic example is \( 1^\infty \), which we encounter with the function \( f(x) = \left(1 - \frac{2}{x}\right)^x \) as \( x \to \infty \). When \( x \) is very large, \( \frac{2}{x} \) becomes close to zero, and \( 1 - \frac{2}{x} \) approaches 1. Raising 1 to the power of infinity seems like it should simply be 1, but it's actually an indeterminate form.

Why does this happen? Well, this form indicates that a small change multiplier, when raised to the power of an incredibly large number, can significantly impact the outcome. These indeterminate forms often point to the natural exponential function \( e \) being involved, such as in the approximation \( \left(1 + \frac{a}{b}\right)^b \approx e^a \).

Recognizing and interpreting indeterminate forms involves:

• Analyzing the terms individually as the variable tends to infinity or zero.
• Identifying any known patterns or limits, especially those linked to the number \( e \).

This insight often leads us to the correct analytical or numerical solution.

Exponential Functions

Exponential functions play a pivotal role in understanding limits that involve expressions like \( 1^\infty \). The function \( f(x) = \left(1 - \frac{2}{x}\right)^x \) builds upon the idea of exponential growth or decay, depending on the variables involved. Generally, an exponential function can be described by the form \( a^x \), where the base \( a \) is a constant. In calculus, functions resembling \( \left(1 + \frac{a}{b}\right)^b \) as \( b \to \infty \) are linked to the exponential constant \( e \), which is approximately equal to 2.718.

Here's how exponential functions fit in:

• They model exponential growth and decay.
• Key in describing natural phenomena and questions involving continuous compounding.
• Essential when calculating limits that form a base around 1 raised to growing exponents.

Understanding exponential functions provides necessary groundwork to tackle limits that seem complex initially, like those presenting \( 1^\infty \), by revealing deeper linkages to the natural base \( e \).

Table of Values

Using a table of values complements graphical estimation by providing numeric insights into how a function behaves as the variable varies. For our function \( f(x) = \left(1 - \frac{2}{x}\right)^x \), instead of viewing it on a graph, we calculate and record specific values. This concrete data helps verify graph-based predictions, especially when precision to more decimal places is required.

To use a table of values effectively:

• Compute \( f(x) \) for a sequence of increasing \( x \) values, like 50, 100, 150, 200.
• Notice the pattern in values as \( x \) becomes larger.

From the calculated values \( f(50) = 0.1383 \), \( f(100) = 0.1353 \), \( f(150) = 0.1351 \), and \( f(200) = 0.1350 \), we see a consistent trend converging towards a specific number. This step-by-step approach helps clarify the trend accurately, confirming the limit to four decimal places as approximately 0.1353.