Problem 18 - Precalculus: Mathematical for Calculus

Question

Use a table of values to estimate the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow \infty}\left(1+\frac{2}{x}\right)^{3 x}$$

Step-by-Step Solution

Answer

The limit is \( e^6 \).

Step 1: Understand the Problem

We need to estimate the limit \( \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{3x} \). This involves examining how the expression behaves as \( x \) approaches infinity.

Step 2: Create a Table of Values

We will choose several large values of \( x \) to plug into the expression \( \left(1 + \frac{2}{x}\right)^{3x} \) and observe the results. For instance, \( x = 100, 1000, 10000, \) etc.

Step 3: Calculate and Analyze Each Value

Calculate \( \left(1 + \frac{2}{100}\right)^{300} \), \( \left(1 + \frac{2}{1000}\right)^{3000} \), \( \left(1 + \frac{2}{10000}\right)^{30000} \). These values approach a certain number as \( x \) increases.

Step 4: Identify the Limit from the Table

As \( x \) becomes larger, you should notice the values approach a more consistent number. For this expression, the values will approach \( e^6 \) as inferred from the properties of exponential expressions.

Step 5: Graph the Function for Visual Confirmation

Using a graphing calculator or software, plot the function \( y = \left(1 + \frac{2}{x}\right)^{3x} \). Observe that as \( x \) approaches infinity, the graph levels off around a particular value.

Step 6: Confirm the Result with the Graph

The graph should show the function approaching a horizontal asymptote at \( y = e^6 \), confirming the numeric estimation from the table.

Key Concepts

Exponential Functions

Exponential functions are critical in calculus and various applications. The expression \( \left(1 + \frac{2}{x}\right)^{3x} \) is a specific example of an exponential function with a slight twist. Here, as \( x \) increases, the exponent of the function grows very large. However, the base \( 1 + \frac{2}{x} \) approaches 1 as \( x \) becomes infinitely large. This scenario typically suggests the function is approaching a limit that involves the exponential constant \( e \). The intuition lies in the fundamental limit theorem that states \( \left(1 + \frac{a}{n}\right)^n \rightarrow e^a \) as \( n \) approaches infinity. In our problem, \( a = 2 \) and \( n = 3x \), leading the function to approach \( e^6 \) as \( x \to \infty \). Understanding these critical details about exponential functions helps decode such limits successfully.

Table of Values

Creating a table of values is a practical method in calculus to predict behavior of limits. When we plug large values of \( x \) into the expression \( \left(1 + \frac{2}{x}\right)^{3x} \), observing the behavior of the spot value allows us to estimate the limit. By selecting large \( x \) values like 100, 1000, and 10000 and computing their corresponding results, you can see a trend in the results.

• For example, compute \( \left(1 + \frac{2}{100}\right)^{300} \), \( \left(1 + \frac{2}{1000}\right)^{3000} \) and so forth.
• Record your results, noticing how the values converge as \( x \) increases.

This provides a numeric insight into what the expression approaches as \( x \to \infty \), confirming assumptions made with calculus reasoning.

Graphical Analysis

Graphical analysis provides a visual representation of how functions behave, particularly as \( x \to \infty \). Using a graphing calculator or software to plot \( y = \left(1 + \frac{2}{x}\right)^{3x} \) clearly shows the behavior of the function. When you sketch this function for large \( x \) values, the graph starts to flatten and approach a specific horizontal line, known as an asymptote.

• This particular graph will level off around \( y = e^6 \).
• Such an approach confirms numerical estimates from the table of values.

The graphical analysis is incredibly insightful as it offers an empirical way of validating theoretical predictions about limits and provides a clearer understanding of the function's behavior as \( x \) becomes very large.