Problem 39 - Single Variable Calculus

Question

Let \(f(x)=\frac{\sin x}{x}\). a. Show that \(-\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}\), for \(x>0\). b. Use the results of (a) and the Squeeze Theorem (which also holds for limits at infinity) to find \(\lim _{x \rightarrow \infty} \frac{\sin x}{x}\). c. Plot the graphs of \(f(x)=-\frac{1}{x}, g(x)=\frac{\sin x}{x}\), and \(h(x)=\frac{1}{x}\) using the viewing window \([0,20] \times\left[-\frac{1}{2}, \frac{1}{2}\right]\).

Step-by-Step Solution

Answer

a. The inequality \(-\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}\) holds true for \(x > 0\). b. Applying the Squeeze Theorem, \(\lim_{x \rightarrow \infty} \frac{\sin x}{x} = 0\). c. The graph of \(g(x)=\frac{\sin x}{x}\) lies within the bounds of \(f(x)=-\frac{1}{x}\) and \(h(x)=\frac{1}{x}\) in the given viewing window \([0, 20] \times \left[-\frac{1}{2}, \frac{1}{2}\right]\).

Step 1: Prove the inequality

To show the inequality, we start by considering the sine function. Recall that \(-1 \leq \sin x \leq 1\) for all values of \(x\). Dividing this inequality by \(x\) (given \(x > 0\)), we get \(-\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}\). Therefore, the inequality holds true for \(x > 0\).

Step 2: Apply the Squeeze Theorem

Now we want to find the limit of the function \(f(x) = \frac{\sin x}{x}\) as \(x \rightarrow \infty\). From step 1, we know that for \(x > 0\), the inequality \(-\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}\) holds true. We first find the limits of the bounding functions as \(x\) approaches infinity: \(\lim_{x \rightarrow \infty} -\frac{1}{x} = 0\) \(\lim_{x \rightarrow \infty} \frac{1}{x} = 0\) Now applying the Squeeze Theorem, we have: \(\lim_{x \rightarrow \infty} \frac{\sin x}{x} = 0\)

Step 3: Plot the graphs

We want to plot the graphs of functions \(f(x)=-\frac{1}{x}\), \(g(x)=\frac{\sin x}{x}\), and \(h(x)=\frac{1}{x}\) in the provided viewing window, which is \([0, 20] \times \left[-\frac{1}{2}, \frac{1}{2}\right]\). It is noted that plotting these graphs requires graphing software or a calculator, but we can analyze the functions in this window: - The function \(f(x)=-\frac{1}{x}\) is a decreasing function ranging between 0 and -0.5 within the given window. - The function \(g(x)=\frac{\sin x}{x}\) oscillates between the functions \(f(x)\) and \(h(x)\), and the oscillations decrease as \(x\) increases. - The function \(h(x)=\frac{1}{x}\) is an increasing function ranging between 0 and 0.5 within the given window. By using graphing software or a calculator, you should observe that the graph of \(g(x)\) lies within the bounds of \(f(x)\) and \(h(x)\) in the given window.

Key Concepts

Limits at Infinity

Understanding limits at infinity is fundamental in calculus, and it refers to the behavior of a function as the input value grows without bound. Consider a function like \( f(x) = \frac{1}{x} \) as an example. As the value of \( x \) becomes larger and larger (approaching infinity), the fraction gets smaller since the denominator is increasing, and as a result, the value of \( f(x) \) approaches zero.

When we say \( \lim_{x \rightarrow \infty} f(x) = L \) where \( L \) is a real number, we're stating that as \( x \) becomes arbitrarily large, \( f(x) \) gets closer and closer to the value \( L \) and stays close to \( L \) within any predefined distance, no matter how small. This is crucial in understanding the end behavior of functions and is particularly useful when evaluating improper integrals or solving problems involving asymptotic behavior.

Inequalities in Calculus

In the realm of calculus, dealing with inequalities involves understanding how to compare the values of different functions across intervals. This understanding plays a key role in strategies like the Squeeze Theorem, which pinpoints the limit of a function by trapping it between two others whose limits are known.

The Squeeze Theorem relies on bounding a function with two others that are easier to evaluate. For instance, if \( f(x) \) is always greater than \( g(x) \) and less than \( h(x) \) for all \( x \) in an interval and if both \( g(x) \) and \( h(x) \) approach the limit \( L \) as \( x \) approaches infinity, then \( f(x) \) must also approach \( L \) as \( x \) approaches infinity. This method is particularly useful for dealing with functions that are difficult to evaluate using standard limit techniques, allowing us to draw conclusions based on the behavior of simpler, more understandable functions.

Graphing Functions

Graphing functions is a visual way to understand the behavior and properties of functions. It helps us see how functions change as their inputs vary, and it's particularly useful for spotting trends, such as asymptotic behavior, intercepts, and intervals of increase or decrease.

When plotting a function, like \( g(x) = \frac{\sin x}{x} \) in a graphing window, one must consider key features such as x-intercepts, y-intercepts, maxima, minima, and the general shape of the graph. The window \( [0,20] \times [-\frac{1}{2}, \frac{1}{2}] \) allows us to analyze the graph of \( g(x) \) within a specific domain and range. Within this window, we can observe the long-term behavior, noting how \( g(x) \) is bounded by \( f(x) = -\frac{1}{x} \) and \( h(x) = \frac{1}{x} \) as \( x \) approaches infinity. The visual representation assists in understanding limits, continuity, and the overall aspect of the function, which is an essential context for applying the Squeeze Theorem or any other analytical method.