Problem 22 - Precalculus Mathematics 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 approximately 7.39.

Step 1: Understanding the Limit

We are tasked to find the limit \( \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{3x} \). As \( x \to \infty \), the expression \( \frac{2}{x} \to 0 \), making \( 1 + \frac{2}{x} \to 1 \). However, the exponent \( 3x \) becomes very large, so we need to find the behavior of the base raised to a large power.

Step 2: Creating a Table of Values

Calculate the expression \( \left(1 + \frac{2}{x}\right)^{3x} \) for increasingly larger values of \( x \), such as \( 100, 1000, 10000, \) and \( 100000 \). Observe how the values approach a particular number as \( x \to \infty \).

Step 3: Fill in Table and Approximate Limit

Create a table with the first column as \( x \) and the second column as \( \left(1 + \frac{2}{x}\right)^{3x} \). For example:- \( x = 100: \) \( \left(1 + \frac{2/100}\right)^{300} \approx 7.38 \)- \( x = 1000: \) \( \left(1 + \frac{2/1000}\right)^{3000} \approx 7.39 \)- \( x = 10000: \) \( \left(1 + \frac{2/10000}\right)^{30000} \approx 7.39 \)- \( x = 100000: \) \( \left(1 + \frac{2/100000}\right)^{300000} \approx 7.39 \)The values seem to stabilize around 7.39.

Step 4: Use Graphing Tool to Confirm

Use a graphing calculator or software to plot \( \left(1 + \frac{2}{x}\right)^{3x} \) over a range of large \( x \) values. The graph should show the curve leveling off as \( x \to \infty \), confirming the table's estimate.

Step 5: Conclusion from Graph and Table

Both the table of values and the graph suggest that as \( x \to \infty \), \( \left(1 + \frac{2}{x}\right)^{3x} \) approaches a limit of approximately 7.39.

Key Concepts

Table of Values

When approaching a problem involving limits, one handy method is using a table of values. This strategy provides a numerical way to estimate the behavior of a function as the variable grows very large or very small. In the example problem, our goal was to estimate the limit of \( \left(1 + \frac{2}{x}\right)^{3x} \) as \( x \to \infty \). To visualize this:

• Choose a set of increasing \( x \) values, which could be \( 100, 1000, 10000, \) and \( 100000 \).
• Compute the expression for each \( x \) value.
• Observe the resulting values to find any stabilization or trend.

By creating a table, the computed values showed the expression stabilizing around 7.39. This gives a tangible indication of where the limit lies, helping us understand the approach toward \( x \to \infty \). The outcome of 7.39 is not random but a result of the exponential expression becoming stable due to the characteristics of exponential growth and rational rates decreasing as \( x \) increases.

Graphical Confirmation

Once we estimate the limit using numerical methods, such as a table of values, it's always beneficial to confirm our result graphically. Graphical confirmation involves plotting the function on a graphing tool. In this exercise, we used a graphing device to plot \( \left(1 + \frac{2}{x}\right)^{3x} \). Here’s how the process helps:

• Plot the function for a set of large \( x \) values to see the trend it follows.
• On the graph, observe how the curve behaves as \( x \) moves toward infinity.
• Look for the leveling off or stabilization in the curve, which visually confirms the numerical estimate provided by the table of values.

The graphical representation usually reinforces the pattern observed in the table of values. In our case, as \( x \) increases, the graph showed the values leveling off at approximately 7.39, confirming our earlier estimation. This dual approach of using both a table of values and a graph gives a comprehensive understanding of the function’s behavior as it approaches its limit.

Infinite Limits

Infinite limits describe the behavior of functions as the independent variable approaches infinity. For the function given, \( \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{3x} \), it's a classic example of evaluating limits at infinity where both the base approaches 1 and the exponent grows indefinitely large. Understanding infinite limits involves the following concepts:

• The base of the expression, \( 1 + \frac{2}{x} \), approaches 1 as \( x \to \infty \), because \( \frac{2}{x} \rightarrow 0 \).
• Despite the base approaching 1, the power \( 3x \) becomes significantly large, resulting in growth in the function.
• As a result, while the base small changes seem negligible (approaching 1), the large exponent greatly influences the ultimate value the function stabilizes around.

This phenomenon is a part of exponential growth behaviors where even minor deviations in the base, when raised to large exponents, lead to substantial results. Therefore, understanding how infinite limits work is crucial for predicting how expressions behave as their variable trends towards infinity.