Problem 10 - Precalculus with Limits

Question

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \left(\dfrac{4}{3x} - 5 \right) \\]

Step-by-Step Solution

Answer

The limit as \(x\) tends to infinity of the function \(f(x) = \dfrac{4}{3x} - 5\) is 0.

Step 1: Divide Each Term by \(x\)

Divide each term in the function by \(x\). Thus, the function becomes \(\dfrac{4}{3x^2} - \dfrac{5}{x}\)

Step 2: Evaluate the Limit

Now, take the limit as \(x\) tends to infinity of the function. This is done by substituting \(x\) with infinity in the function from step 1. When a constant is divided by infinity, the result is zero. Thus, both terms become zero and the limit is \(0 - 0 = 0\).

Step 3: Use a Graphing Utility

Use a graphing utility to plot the function \(f(x) = \dfrac{4}{3x} - 5\). Observing the plot, one will notice that as \(x\) increases, the function approaches \(0\). This means that the graphical behavior of the function confirms the analytical finding from Step 2.

Key Concepts

Understanding Infinity in Limits

Infinity often appears in calculus, especially in the context of limits. It's an abstract concept that represents something without any bound or end. When we say \(x \rightarrow \infty\), it means that \(x\) is increasing beyond all finite bounds. Similarly, \(x \rightarrow -\infty\) indicates \(x\) decreasing without bound.

When evaluating limits as \(x\) approaches infinity, remember:

• If a constant is divided by \(x\), and \(x\) tends to infinity, the result approaches zero. For example, \(\dfrac{a}{x} \rightarrow 0\) as \(x \rightarrow \infty\).
• Infinity is not a number; it's a concept. Therefore, it cannot be used in standard arithmetic operations.

Understanding these key points helps in evaluating the behavior of functions as they extend indefinitely.

Using Graphing Utilities

A graphing utility is a powerful tool to visualize functions and verify analytical results. These utilities can be software applications or physical calculators that render graphical representations of equations, showing their behavior over a range of values.

To use a graphing utility effectively:

• Enter the function exactly as it appears, for example, \(f(x) = \dfrac{4}{3x} - 5\).
• Set the appropriate window or range for \(x\) to see the graph's behavior as \(x\) increases toward infinity.
• Observe patterns, such as horizontal asymptotes, which indicate the value the function is approaching.

Graphing utilities not only support calculations but also provide a visual confirmation of limit behavior, making abstract concepts more tangible.

Evaluating Limits

Evaluating limits is a fundamental task in calculus used to understand the behavior of functions as they approach a certain point, including infinity. The process involves:

• Manipulating the function to a simpler form if possible, such as dividing each term by \(x\).
• Substituting the value \(x\) is approaching into the function. For limits at infinity, consider what happens to each term separately.
• Applying special rules or theorems, such as L'Hopital's Rule for indeterminate forms.

These steps allow you to analytically determine limits and understand how functions behave in extreme conditions. Confirming results with a graphing utility ensures accuracy and provides a complete picture of the function's behavior.