Problem 38 - Calculus of a Single Variable

Question

Finding a Limit In Exercises \(19-38,\) find the limit. $$ \lim _{x \rightarrow \infty} \frac{x-\cos x}{x} $$

Step-by-Step Solution

Answer

The limit as x approaches infinity for \( \frac{x-\cos x}{x} \) is 1.

Step 1: Separate the Fraction

We can separate the fraction in the equation to simplify it. The equation will then be split into two separate limits: \[\lim _{x\rightarrow \infty } x/x - \lim _{x\rightarrow \infty } \cos x /x\] By simplifying, we get: \[\lim _{x\rightarrow \infty } 1 - \lim _{x\rightarrow \infty } \cos x /x \]

Step 2: Evaluate the Limits Separately

First term \(\lim _{x\rightarrow \infty } 1\) is just 1 no matter what the value of x is. We will treat the second limit separately, \(\lim _{x\rightarrow \infty } cos x /x .\) As x approaches infinity, \(\cos x\) will always oscillate between -1 and 1. But x approaches a certain large number, the fraction will then gradually approach the number 0.

Step 3: Add the Results

After evaluating both limits, we just need to subtract the second from the first: \[ 1 - 0 = 1 \]

Key Concepts

Understanding Infinite Limits

When we're dealing with infinite limits, we're looking at what happens as a variable, like \(x\), increases or decreases without bound. Essentially, we explore the behavior of a function as its input becomes very large or very small.
For example, in the exercise, \(\lim_{x \to \infty}\) examines what happens as \(x\) grows larger and larger. This allows us to predict the end behavior of functions, simplifying complex expressions, and understanding long-term trends. Such limits often highlight the dominant terms, both in growth and decay, and can reveal insights into how different parts factor into the overall behavior of the function.

The Role of Oscillating Functions

Oscillating functions, like \(\cos x\), are ones that repeatedly move between certain values—in this case, between -1 and 1. When evaluating limits involving oscillating functions, especially as \(x\) approaches infinity, it's crucial to understand their behavior.
Since \(\cos x\) does not settle towards a specific number and keeps swinging between its range, it presents an interesting scenario. When combined in a fraction with x, this oscillation means that as \(x\) gets very large, the effect of \(\cos x\) diminishes because it is divided by an infinitely growing number, ultimately trending the fraction towards zero.

Simplifying with Fraction Separation

Fraction separation is a useful technique when working with complex fractions. It involves breaking a single fraction into separate, simpler parts to make evaluating limits easier.
For instance, \(\frac{x - \cos x}{x}\) can be rewritten as \(\frac{x}{x} - \frac{\cos x}{x}\). This separation allows each part to be analyzed on its own:

• \(\frac{x}{x} = 1\), which is straightforward and constant as \(x\) grows.
• \(\frac{\cos x}{x}\) highlights the diminishing impact of \(\cos x\), simplifying the overall approach.

By separating the terms, you simplify the evaluation process significantly, making it easier to handle individual components.

Evaluating Limits Like a Pro

Evaluating limits involves determining what value a function approaches as the variable within it approaches a specific point. It may sound complex, but with practice, it becomes intuitive.
This exercise involved evaluating two strips of separated fractions:

• The constant \(\lim_{x \to \infty} 1 = 1\), which remains unaffected by \(x\).
• For \(\lim_{x \to \infty} \frac{\cos x}{x}\), as discussed, oscillation takes a backseat to the ever-increasing denominator, leading to a trend towards zero.

Combining these evaluated limits, you end up subtracting zero from one, yielding a clear solution of 1. Evaluating limits like this requires breaking down problems, observing behaviors, and recognizing trends.