Problem 53 - Calculus Early Transcendentals: Pearson New International Edition

Question

Many software packages have programs for calculating limits, although you should be warned that they are not infallible. To develop confidence in your program, use it to recalculate some of the limits in Problems 1-28. Then for each of the following, find the limit or state that it does not exist. $$ \lim _{x \rightarrow 0} \cos (1 / x) $$

Step-by-Step Solution

Answer

The limit does not exist because \(\cos(1/x)\) oscillates indefinitely as \(x\) approaches 0.

Step 1: Understanding the problem

The problem asks us to find the limit of the function \( \cos(1/x) \) as \( x \) approaches 0. To solve this, we need to examine the behavior of the function as \( x \) gets closer to 0.

Step 2: Analyzing the function

The function \( \cos(1/x) \) involves a trigonometric function applied to a reciprocal of \( x \). As \( x \rightarrow 0 \), the value of \( 1/x \) becomes very large, oscillating between positive and negative infinity.

Step 3: Behavior of cosine function

Cosine is a periodic function with a period of \( 2\pi \), which oscillates between -1 and 1. As \( 1/x \) approaches positive or negative infinity, \( \cos(1/x) \) will continue to oscillate rapidly between these two bounds.

Step 4: Conclusion on the limit

Due to the rapid oscillation of \( \cos(1/x) \) without settling to a particular value, the limit does not exist. The function does not approach any single value as \( x \rightarrow 0 \).

Key Concepts

Trigonometric Functions

Trigonometric functions, like cosine, are vital in calculus. They are periodic and often repeat specific values across their intervals. For cosine, the function

• oscillates between -1 and 1,
• has a period of \(2\pi\).

As you explore these functions, it’s essential to understand their behavior over different domains. In this problem, we are focusing on \(\cos(1/x)\). The argument \(1/x\) alters the frequency, leading cosine to quickly change as \(x\) nears 0. Due to the oscillating nature of the cosine function, it can never settle on a single limit as x approaches a specific value. Recognizing how trigonometric functions behave enables you to predict how they might act in limits, which is crucial when dealing with more complex scenarios.Moreover, the reciprocation of \(x\) into \(1/x\) means every small change near x causes significant shifts in input value due to the large output it corresponds to after substituting into the cosine function. This makes analyzing the limit particularly challenging. Understanding these characteristics of trigonometric functions and how they interact with calculus concepts like limits is crucial for mastering these topics.

Limit Behavior

Understanding limit behavior is a core part of calculus. Limits allow us to analyze what a function does as its input approaches a particular value. For a function \(f(x)\), the limit as \(x\) approaches a point \(a\) examines what value \(f(x)\) gets very close to as \(x\) nears \(a\).In this problem, as \(x\) approaches 0, the challenge is that the function \(\cos(1/x)\) does not slowly inch toward a number. Instead, its behavior is erratic. Therefore, a major takeaway here is recognizing when a limit might not exist due to such unpredictable nature.Not all trigonometric limits behave nicely. Some limits, like fractions involving polynomials, might simplify gradually to a point. But trig functions can misbehave greatly, especially when incorporating infinite elements, like \(1/x\). This is why the function \(\cos(1/x)\) fails to have a definable limit at 0 as it endlessly jumps around its bounded range.

Oscillation in Calculus

Oscillation refers to a function repeatedly changing between two states without settling into a point as an input approaches a given number. In calculus, this is especially significant in examining limits. For the function \(\cos(1/x)\), each time \(x\) gets smaller, \(1/x\) becomes larger in magnitude, causing the cosine to switch rapidly between its bounds, -1 and 1.

• This process is called oscillation, and
• Oscillation prevents the function from settling on a single limit.

This means, as \(x\rightarrow0\), the function never decides on a consistent output but goes back and forth endlessly. While the specific path of values changes continuously, the amplitude remains fixed with no resolution towards a specific number.A clear understanding of oscillation helps discern when limits won’t exist. Recognizing this behavior in problems will inform your approach, as you realize that the function cannot reach a smooth conclusion due to its wild swings. Therefore, knowing when a function oscillates can save you time by immediately recognizing why certain limits do not exist.