Problem 19 - Calculus for Biology and Medicine

Question

Let \(f(x)=\frac{\ln x}{x}, \quad x>0\) (a) Use a graphing calculator to graph \(y=f(x)\). (b) Use a graphing calculator to investigate the values of \(x\) for which $$\frac{1}{x} \leq \frac{\ln x}{x} \leq \frac{1}{\sqrt{x}}$$ holds. (c) Use your result in (b) to explain why: \(\lim _{x \rightarrow \infty} \frac{\ln x}{x}=0\).

Step-by-Step Solution

Answer

(a) Use a calculator to graph. (b) Graph reveals \( x \) range meeting inequalities. (c) \( \lim_{x\rightarrow\infty}\frac{\ln x}{x}=0 \) by Squeeze Theorem.

Step 1: Understand the Function

The given function is \( f(x) = \frac{\ln x}{x} \) where \( x > 0 \). Our goal is to analyze its behavior, visualize its graph, and determine bounds for \( x \) such that \( \frac{1}{x} \leq \frac{\ln x}{x} \leq \frac{1}{\sqrt{x}} \).

Step 2: Graph the Function with a Calculator

Using a graphing calculator, plot the function \( y = \frac{\ln x}{x} \). The graph typically starts at the origin, increases, reaches a peak, and then gradually decreases towards zero as \( x \) increases.

Step 3: Establish Inequality Bounds

We need to investigate when \( \frac{1}{x} \leq \frac{\ln x}{x} \leq \frac{1}{\sqrt{x}} \) holds true. Start by plotting \( y = \frac{1}{x} \) and \( y = \frac{1}{\sqrt{x}} \) on the same graph using the calculator to visualize where \( f(x) \) fits within these bounds.

Step 4: Analyze Intersection Points

Calculate where \( \frac{\ln x}{x} \) intersects with \( \frac{1}{x} \) and \( \frac{1}{\sqrt{x}} \). By comparing graphs, you'll see a certain range of \( x \) where both inequalities are satisfied. This requires numerical estimation with the calculator.

Step 5: Interpret Graphical Results

From the graph and intersections, observe the portion of the \( x \)-axis where \( \frac{1}{x} \) is below \( \frac{\ln x}{x} \) and above \( \frac{1}{\sqrt{x}} \). Confirm these values as the conditions hold for a specific \( x \) range.

Step 6: Conclusion of Limits

As \( x \rightarrow \infty \), both bounding functions \( \frac{1}{x} \) and \( \frac{1}{\sqrt{x}} \) approach zero. Since \( \frac{\ln x}{x} \) is squeezed between them, it also approaches zero due to the Squeeze Theorem.

Key Concepts

limit

The concept of a limit is one of the foundational principles in calculus. It describes the value that a function approaches as the input approaches a certain point. In this exercise, we are looking at the limit of the function \[ f(x) = \frac{\ln x}{x} \]as \( x \) approaches infinity. As \( x \) increases, the natural logarithm \( \ln x \) grows, but the denominator \( x \) grows even faster. This causes the function value to decrease towards zero. Understanding limits help us determine how functions behave over time or under certain conditions. It's like predicting future outcomes in data. Here, the goal was to show that \[ \lim_{x \rightarrow \infty} \frac{\ln x}{x} = 0 \]by examining the behavior of the function's values as the input becomes infinitely large.

graphing calculator

When working with complex functions, a graphing calculator becomes an invaluable tool. It allows us to visualize the function's behavior quickly. For this problem, plotting \( y = \frac{\ln x}{x} \) gives a clearer picture of how the function behaves across the domain of interest. The graph typically shows the function starting at zero, increasing to a peak, and then decreasing towards zero as \( x \) becomes very large. By using a graphing calculator, you can also plot additional functions such as \( y = \frac{1}{x} \) and \( y = \frac{1}{\sqrt{x}} \). This enables you to investigate inequalities and observe how these functions interact with one another. Intersection points and relative positions of the graphs can provide insights into inequalities and help in understanding when the initial function fits within a particular range.

inequalities

In this exercise, we examined the inequalities \[ \frac{1}{x} \leq \frac{\ln x}{x} \leq \frac{1}{\sqrt{x}} \]Inequalities allow us to establish bounds on the possible values that a function can take. By finding these values, we can better understand the behavior and constraints of the function.- **Lower Bound:** The inequality \( \frac{1}{x} \leq \frac{\ln x}{x} \) suggests that the function is always greater than or equal to \( \frac{1}{x} \) for the specified \( x \) values.- **Upper Bound:** Similarly, \( \frac{\ln x}{x} \leq \frac{1}{\sqrt{x}} \) indicates that the function can never exceed \( \frac{1}{\sqrt{x}} \) in this range.By plotting these beside our original function in a graphing calculator, we visually confirm the intervals that satisfy both inequalities. This aspect is crucial in demonstrating how functions converge or diverge under certain conditions.

Squeeze Theorem

The Squeeze Theorem is a powerful tool in calculus that helps establish the limit of a function. It's especially useful when a function is difficult to evaluate directly. The theorem states that if a function is "squeezed" between two others, and if these boundary functions have the same limit at a point, then the squeezed function must also have that limit.For this function \( \frac{\ln x}{x} \) as \( x \rightarrow \infty \), the Squeeze Theorem plays a crucial role:- The function is squeezed between two simpler functions, \( \frac{1}{x} \) and \( \frac{1}{\sqrt{x}} \).- As \( x \rightarrow \infty \), both \( \frac{1}{x} \) and \( \frac{1}{\sqrt{x}} \) tend to zero.Given these boundaries, the Squeeze Theorem assures us that \( \frac{\ln x}{x} \) also approaches this same value — zero. This use of the theorem provides a systematic way to conclude regarding the behavior of more complex functions, using simpler approximations.