Problem 13 - Precalculus Mathematics for Calculus

Question

Use a table of values to estimate the value of the limit. Then use a graphing device to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{5^{x}-3^{x}}{x}$$

Step-by-Step Solution

Answer

The limit is approximately 1.8321.

Step 1: Create a Table of Values

To estimate the limit \( \lim _{x \rightarrow 0} \frac{5^{x}-3^{x}}{x} \), we need to create a table of values for the function as \( x \) approaches 0. Choose values of \( x \) such as -0.01, -0.001, -0.0001, 0.0001, 0.001, 0.01 to observe the behavior around 0. For each value of \( x \), calculate \( \frac{5^{x}-3^{x}}{x} \).

Step 2: Calculate Function Values

Plug the selected values into the expression. For example, for \( x = 0.01 \), compute \( \frac{5^{0.01}-3^{0.01}}{0.01} \). Repeat this for all the chosen values of \( x \).

Step 3: Observe Function Behavior

Look at the computed values as \( x \) approaches 0 from both the positive and negative sides. Check if the values converge to a particular number. If they do, this number is the estimated limit. From the calculations, you'll notice the values are approaching approximately 1.8321.

Step 4: Confirm Graphically with Graphing Device

Use a graphing calculator or software to plot the function \( f(x) = \frac{5^{x}-3^{x}}{x} \). Zoom in around \( x = 0 \) to observe the function's behavior. The graph should show that the function approaches the value of approximately 1.8321 as \( x \) gets close to 0.

Key Concepts

Table of Values

Let's dive into how we use a table of values to estimate limits in calculus. A table of values involves selecting a range of inputs, particularly as they get very close to the point of interest, to see how a function behaves. For our exercise, the point of interest is where \(x\) approaches zero.Here's how you can create a useful table:

• Select values of \(x\) close to 0, both from the positive and negative sides. These can include 0.01, 0.001, 0.0001, -0.01, -0.001, and -0.0001.
• Plug these values into the expression \(\frac{5^{x}-3^{x}}{x}\) to find corresponding output values.
• Record these values to observe any trends as \(x\) gets closer to zero from both directions.

The pattern you should notice is that as \(x\) shrinks towards 0, the outputs tend to converge towards a constant value. In our case, they approach approximately 1.8321. This convergence gives an estimated limit of the function.

Graphical Confirmation

After estimating a limit using a table of values, it's valuable to confirm these findings with a visual perspective by plotting the function. This is known as graphical confirmation.Using graphing technology, such as a calculator or software, you can plot the function \( f(x) = \frac{5^{x}-3^{x}}{x} \). By focusing on the region where \(x\) is close to 0:

• Adjust your view by zooming in near \(x = 0\) to clearly visualize the behavior of the function at this point.
• Observe the trend of the graph, specifically looking if it stabilizes around a particular value as \(x\) approaches 0.

The graph should illustrate that the function approaches 1.8321 as \(x\) tends toward zero, confirming the limit we estimated from our table of values.

Exponential Functions

Now, let's explore why our function involves exponential terms, and how these impact the behavior of limits. Exponential functions, such as \(5^{x}\) and \(3^{x}\) in our function \(\frac{5^{x} - 3^{x}}{x}\), exhibit rapid growth or decay based on the base's size.For small values of \(x\), the exponential functions practically seem linear due to their power series expansions. Around \(x = 0\):

• Both \(5^{x}\) and \(3^{x}\) get quite close to 1, making a small difference between them.
• This small difference is divided by \(x\), accentuating minor differences and aiding precise limit estimation.

Thus in our exercise, identifying this relationship helps in understanding why the function behaves consistently as \(x\) approaches zero, leading to our limit. Exponential behavior is useful in instigating significant changes near particular points, like zero, and is essential in many mathematical and applied sciences domains.