Problem 40 - Calculus Early Transcendentals

Question

Limits by graphing Use the zoom and trace features of a graphing utility to approximate the following limits. $$\lim _{x \rightarrow 1} \frac{18(\sqrt[3]{x}-1)}{x^{3}-1}$$

Step-by-Step Solution

Answer

Question: Approximate the limit of the function $$\lim _{x \rightarrow 1} \frac{18(\sqrt[3]{x}-1)}{x^{3}-1}$$ using the zoom and trace features of a graphing utility. Answer: Based on the graphing analysis, we can approximate the limit as: $$\lim _{x \rightarrow 1} \frac{18(\sqrt[3]{x}-1)}{x^{3}-1} \approx 6$$.

Step 1: Graph the function

First, input the function into a graphing calculator or graphing software like Desmos or GeoGebra. The function is: $$f(x) = \frac{18(\sqrt[3]{x}-1)}{x^{3}-1}$$.

Step 2: Zoom in and trace

Zoom in on the graph of the function as x approaches 1 from both the left and the right. To do this, keep track of the x and y coordinates in the trace function. You can continue to zoom in until the y-coordinate stabilizes, providing an approximation of the limit.

Step 3: Approximate the limit

Keep note of the stable y-value you observed when x approached 1. This will be your approximation for the limit. It should be close to the actual limit of the function, which is about 6. Therefore, based on the graphing analysis, we can approximate the limit as: $$\lim _{x \rightarrow 1} \frac{18(\sqrt[3]{x}-1)}{x^{3}-1} \approx 6$$.

Key Concepts

Limit of a Function

A limit is a fundamental concept in calculus that describes the behavior of a function as its input (or 'x' value) approaches a certain point. The notation
\(\lim_{x \to a} f(x)\) represents the value that the function \(f(x)\) gets closer to as \(x\) approaches \(a\). Limits are essential for understanding continuity, derivatives, and integrals. When evaluating limits, the function doesn't always need to equal its limit at the point. For example, limits can exist even if there's a hole or asymptote at \(x = a\).

Graphing Calculator Techniques

Graphing calculators are powerful tools for visualizing mathematical concepts, which can be particularly helpful when studying limits. To analyze limits using a graphing calculator, you first need to input the given function into the calculator.

After inputting the function, plot the graph to visualize the behavior of the function. Graphing calculators often come with a variety of functions such as zooming and tracing, which allow for more detailed analysis of the function's graph. By examining the graph, you can gain insights into the behavior of the function around the point of interest for the limit.

Zoom and Trace Features

The zoom and trace features on graphing calculators are indispensable for exploring the details of graphs. Zooming in on a graph helps you examine the behavior of a function at a particular region, which is crucial when approximating limits.

For example, you can zoom in on the area near \(x = 1\) to see how the function behaves as it approaches that point. The trace feature allows you to move along the graph and observe the corresponding coordinates. By tracing the function, you can pinpoint the values of \(y\) as \(x\) gets closer to the specified value, which in this case helps to approximate the limit of the function as \(x\) approaches 1.

Graphical Limit Approximation

To approximate a limit graphically, use the graphing calculator to draw the function and then utilize the zoom and trace features. As you repeatedly zoom in on the point where \(x\) approaches the value of interest, the graph's behavior will reveal the limit's approximation. The y-coordinate where the x-value is near the point of interest will become more stable. You’ve found a good approximation when further zooming doesn’t change the y-value significantly.

In the given exercise, by tracing the function as \(x\) approaches 1 and observing the stabilizing y-values, you can conclude the limit to be approximately 6. Remember, this method provides an estimate that should be verified analytically to ensure precision.