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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.

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.

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.

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.