Problem 23 - Calculus Early Transcendentals

Question

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to confirm your result graphically. $$\lim _{x \rightarrow 1} \frac{x^{6}-1}{x^{10}-1}$$

Step-by-Step Solution

Answer

The limit is approximately 0.9.

Step 1: Set Up the Table of Values

Create a table of values for \( x \) approaching 1 from both the left and right. Choose values close to but not equal to 1, such as 0.9, 0.99, 1.01, and 1.1. For each \( x \), compute the expression \( \frac{x^6 - 1}{x^{10} - 1} \).

Step 2: Compute the Values

Calculate the expression for \( x = 0.9, 0.99, 1.01, 1.1 \):- At \( x = 0.9 \), \( \frac{0.9^6 - 1}{0.9^{10} - 1} = \frac{0.531441 - 1}{0.348678 - 1} \approx 0.8234\) - At \( x = 0.99 \), \( \frac{0.99^6 - 1}{0.99^{10} - 1} \approx 0.8971 \)- At \( x = 1.01 \), \( \frac{1.01^6 - 1}{1.01^{10} - 1} \approx 0.9040 \)- At \( x = 1.1 \), \( \frac{1.1^6 - 1}{1.1^{10} - 1} \approx 0.8384 \)

Step 3: Analyze the Values

The values of the expression as \( x \) approaches 1 are getting closer to approximately 0.9 from both sides. This suggests the limit is around 0.9.

Step 4: Confirm with a Graph

Use a graphing utility to plot \( y = \frac{x^6 - 1}{x^{10} - 1} \). Observe the behavior of the graph as \( x \) approaches 1. The graph should show the value approaching around 0.9 as \( x \to 1 \).

Key Concepts

Table of Values

Using a table of values is a practical method for estimating limits. This approach involves choosing values of the variable that are close to the point of interest, and then evaluating the given expression at these points. In this exercise, we are interested in what happens as \( x \) approaches 1. To capture values on either side of 1, select numbers slightly less than and slightly greater than 1.

• Example numbers: 0.9, 0.99, 1.01, 1.1.

Once you have selected these values, substitute them into the expression \( \frac{x^6 - 1}{x^{10} - 1} \). This will allow you to calculate an approximation of the output as \( x \) nears 1. Keep track of these outputs to see how they change as \( x \) approaches 1 from either side. The idea is that your calculated outputs will converge to a particular number, giving an estimate for the limit.

Approaching Limits

Approaching limits involves observing the behavior of a function as the input value gets infinitely close to a point of interest. In our exercise, we focus on how the expression \( \frac{x^6 - 1}{x^{10} - 1} \) behaves as \( x \) nears 1. This is crucial because the limit essentially tells us the value the function is tending towards.By examining the computed values in the table, notice that as \( x \) gets closer to 1 from both the left (0.9, 0.99) and right (1.01, 1.1), the resulting values of the expression start to settle around a particular number—in this case, approximately 0.9. This pattern of narrowing values indicates that the limit of the function as \( x \) approaches 1 is about 0.9.Approaching limits isn't just about numbers on a page; it's a vital process in understanding the continuity and behavior of functions.

Graphical Confirmation

Graphical confirmation is a powerful way to verify the results obtained from numerical estimations of limits. By using a graphing device or software, you can visualize the function's behavior. In this exercise, plotting the expression \( y = \frac{x^6 - 1}{x^{10} - 1} \) gives you a visual representation of how the function acts as \( x \) approaches 1.To plot, input the function into a graphing tool and focus on the area around \( x = 1 \). Observe the direction the graph takes as it nears this point. If the plot shows the line leveling out towards 0.9, this visually confirms your calculated estimate. Graphical confirmation not only reaffirms the numerical approximation but also enhances understanding by visual insight, making the concept of limits more tangible and intuitive.