Problem 65 - Calculus: Early Transcendentals

Question

(a) Use the Squeeze Theorem to evaluate \( \displaystyle \lim_{x \to \infty} \frac{\sin x}{x} \). (b) Graph \( f(x) = (\sin x)/x \). How many times does the graph cross the asymptote?

Step-by-Step Solution

Answer

(a) \( \lim_{x \to \infty} \frac{\sin x}{x} = 0 \). (b) The graph crosses the asymptote infinitely many times.

Step 1: Understand the Squeeze Theorem

The Squeeze Theorem states that if you have three functions \( g(x), f(x), \) and \( h(x) \) such that \( g(x) \leq f(x) \leq h(x) \) for all \( x \) in some interval, except possibly at a point \( c \), and \( \lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L \), then \( \lim_{x \to c} f(x) = L \).

Step 2: Apply the Squeeze Theorem to \( \frac{\sin x}{x} \)

We know that \(-1 \leq \sin x \leq 1\). Therefore, we can express:\[ -\frac{1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x} \]As \( x \to \infty \), both \(-\frac{1}{x}\) and \(\frac{1}{x}\) approach 0. By the Squeeze Theorem, we then infer that:\[ \lim_{x \to \infty} \frac{\sin x}{x} = 0 \]

Step 3: Understand Graphing and Asymptotes

The function \( f(x) = \frac{\sin x}{x} \) has a horizontal asymptote of \( y = 0 \) as determined in Step 2. We will graph the function to see how it behaves and determine how many times it crosses \( y = 0 \).

Step 4: Graph \( f(x) = \frac{\sin x}{x} \)

When graphing, you will notice that the function \( \frac{\sin x}{x} \) oscillates because \(\sin x\) oscillates between -1 and 1. As \(x\) increases, the amplitude of these oscillations decreases since \(x\) in the denominator makes the function approach zero. Thus, it crosses the horizontal asymptote \(y = 0\) infinitely often as this oscillation happens for each crossing of the x-axis by \(\sin x\).

Key Concepts

Limit

When we discuss limits in calculus, we're talking about what a function approaches as the input approaches a certain value. Limits are essential for understanding the behavior of functions as they get closer to a specific point or when inputs become very large or very small. In the context of the Squeeze Theorem, limits help us determine the behavior of complex functions that are difficult to evaluate directly.

• The limit notation \( \lim_{x \to c} f(x) = L \) indicates that as \( x \) gets closer to \( c \), \( f(x) \) approaches \( L \).
• The Squeeze Theorem uses limits to find \( L \) when a function is "squeezed" between two other functions whose limits are known.

In the given problem, using the Squeeze Theorem, we show that:\(\lim_{x \to \infty} \frac{\sin x}{x} = 0\) as both bounding functions approach zero,demonstrating the power of the limit in solving otherwise complex problems.

Asymptote

Asymptotes represent a line that a graph of a function approaches but never actually reaches. They are mainly used to describe the behavior of graphs as they extend towards infinity or negative infinity. Understanding asymptotes is fundamental when analyzing functions like \( f(x) = \frac{\sin x}{x} \), as they provide insight into the long-term behavior of the graph.For \( \frac{\sin x}{x} \), there's a horizontal asymptote at \( y = 0 \). This implies:

• As \( x \to \infty \), the function approaches zero, flattening out along the horizontal line \( y = 0 \).
• The function will never truly reach or stay at \( y = 0 \), but rather oscillate around it.

Recognizing asymptotes helps in predicting and understanding the behavior of functions, especially those involving infinite limits or domains.

Graphing Sinusoidal Functions

Graphing sinusoidal functions involves translating the oscillatory behavior of sine or cosine functions onto a graph. Since \( \sin x \) repeatedly oscillates between -1 and 1, any function incorporating \( \sin x \), like \( \frac{\sin x}{x} \), will also exhibit a wave-like behavior.

• To graph \( \frac{\sin x}{x} \): note that large values of \( x \) in the denominator cause the amplitude of oscillations to decrease.
• As \( x \) grows, the function approaches the x-axis due to this diminishing amplitude.

This oscillation means the graph crosses the axis many times, a characteristic captured betterthrough visualization, reflecting the underlying "wave" properties of the function modified by the denominator.

Infinite Oscillations

Infinite oscillations occur when a function continues to fluctuate between values without settling to a point as the input becomes very large. This is a common trait when functions involve periodic elements, such as sine or cosine.In the example of \( \frac{\sin x}{x} \):

• The function oscillates infinitely as \( \sin x \) runs through all its cycles indefinitely.
• These oscillations decrease in size as \( x \to \infty \), tightening around the x-axis due to the damping effect of the \( \frac{1}{x} \).

Despite this seemingly erratic behavior, infinite oscillations are predictable in nature, providing a fascinating insight into how periodic functions transform when combined with other mathematical expressions. Recognizing this concept is crucial for a deeper understanding of how combined functions behave at large scales.