Problem 42 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{x \rightarrow \infty} \frac{\sin x}{x} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the limit definition

The problem asks us to evaluate \(\ \lim _{x \rightarrow \infty} \frac{\sin x}{x} \). Our goal is to determine what value \( \frac{\sin x}{x} \) approaches as \( x \) becomes infinitely large.

Step 2: Analyze the function behavior

The function \( \sin x \) oscillates between -1 and 1 for all real values of \( x \). Thus, the numerator of our function, \( \sin x \), is always bounded between -1 and 1.

Step 3: Set up an inequality

Since \( \sin x \) is bounded between -1 and 1, we have:\(\frac{-1}{x} \leq \frac{\sin x}{x} \leq \frac{1}{x}\)for all \( x eq 0 \). This inequality can help us assess the limit behavior of \( \frac{\sin x}{x} \) as \( x \rightarrow \infty \).

Step 4: Apply the Squeeze Theorem

As \( x \rightarrow \infty \), both terms \( \frac{-1}{x} \) and \( \frac{1}{x} \) approach 0, since both are fractions with a constant numerator and a denominator that grows to infinity. Because \( \frac{\sin x}{x} \) is squeezed between these two terms, the Squeeze Theorem tells us that:\[\lim _{x \rightarrow \infty} \frac{\sin x}{x} = 0.\]Thus, the limit of \( \frac{\sin x}{x} \) as \( x \) goes to infinity is 0.

Key Concepts

Limit Theorems

Understanding the different limit theorems is crucial when dealing with calculus problems involving limits. The limit theorems provide us with tools and techniques to calculate or estimate limits of various functions effectively. Some of the most common theorems include:

• Limits of sums: The limit of a sum is the sum of the limits.
• Limits of products: The limit of a product is the product of the limits.
• Limits of quotients: The limit of a quotient is the quotient of the limits, as long as the denominator does not approach zero.
• Squeeze Theorem: If a function is confined between two other functions that have the same limit at a particular point, then the function will also have that limit.

The Squeeze Theorem was utilized effectively in the exercise above. It helped simplify the problem by bounding the function \( \frac{\sin x}{x} \) between \( \frac{-1}{x} \) and \( \frac{1}{x} \). Both of these bounding functions approach zero as \( x \rightarrow \infty \), allowing us to conclude that our target function must also approach zero.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in calculus and pre-calculus. The sine function specifically, \( \sin x \), cycles or oscillates between -1 and 1 indefinitely as \( x \) increases. This characteristic makes sine a

• periodic function,
• bounded, and
• non-monotonic, meaning it does not only increase or decrease but does both.

In the limit problem, understanding the behavior of \( \sin x \) was crucial. The key is that, no matter what value \( x \) takes, \( \sin x \) will be somewhere between -1 and 1. This bounded nature allowed the use of the Squeeze Theorem. Even though \( x \rightarrow \infty \) means \( x \) continues to grow larger, \( \sin x \) does not blow up to infinity. Instead, it stays oscillating between -1 and 1.

Infinite Limits

Infinite limits describe the behavior of a function as the input, such as \( x \), approaches positive or negative infinity. When evaluating infinite limits, it's crucial to determine how the function behaves as the values become exceedingly large or small. In cases like \( \lim _{x \rightarrow \infty} \frac{\sin x}{x} \), we're interested in what happens to the fraction as \( x \) grows indefinitely.One helpful aspect of this type of problem is realizing how growing denominators impact fractions. As \( x \) becomes infinitely large, any constant numerator divided by \( x \) results in the fraction approaching zero. This was evident in the problem where both bounding sides \( \frac{-1}{x} \) and \( \frac{1}{x} \) went to zero, effectively pulling \( \frac{\sin x}{x} \) along with them via the Squeeze Theorem. Infinite limits offer a pathway to understanding function behaviors as inputs expand beyond typical bounds, providing insights into asymptotic tendencies and function stability over extensive domains.