Problem 28 - Essential Calculus

Question

In Exercises \(15-36,\) find the limit. $$ \lim _{x \rightarrow \infty} \cos \frac{1}{x} $$

Step-by-Step Solution

Answer

The limit as \(x\) approaches infinity of \(\cos \frac{1}{x}\) is 1

Step 1: Identify the limit direction

First, we need to find what value \(\frac{1}{x}\) approaches as \(x\) approaches infinity. As \(x\) gets larger and approaches infinity, we know that any number divided by infinity tends to approach zero, hence \(\frac{1}{x}\) approaches 0 as \(x\) tends to infinity.

Step 2: Apply the limit to the function

Once we know that \(\frac{1}{x}\) approaches 0, we can substitute this into the cosine function, as the question asks for \(\lim _{x \rightarrow \infty} \cos \frac{1}{x}\), so when \(\frac{1}{x}\) approaches 0, the inside of the cosine function, or \(\cos \frac{1}{x}\), approaches cosine of 0 as \(x\) tends to infinity.

Step 3: Evaluate the limit using the cosine function.

Now we simply evaluate what \(\cos(0)\) is. We know that the cosine of 0 is 1, so this means that \(\cos \frac{1}{x}\) approaches 1 as \(x\) goes to infinity.

Key Concepts

Cosine function

The cosine function is a fundamental concept from trigonometry, commonly represented as \( \cos(x) \). It is a periodic function and one of the basic trigonometric functions alongside sine and tangent. Cosine relates to the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

• Cosine is an even function, meaning \( \cos(-x) = \cos(x) \).
• It has a range of \([-1, 1]\), indicating that for any input, the output values will always lie between -1 and 1.
• The cosine function is periodic with a period of \(2\pi\), meaning that its values repeat every \(2\pi\) units.

When evaluating the cosine of any angle, like in the exercise where we evaluate \( \cos \frac{1}{x} \) as \( x \) approaches infinity, we need to consider how \( \frac{1}{x} \) changes. Here, understanding the cosine value as a function of dynamic inputs, such as \( \frac{1}{x} \to 0 \), is crucial.Evaluating cosine at \(0\) tells us that \( \cos(0) = 1 \). So, for inputs approaching zero, like \( \frac{1}{x} \) approaching zero as \( x \to \infty \), the cosine function yields an output of 1.

Infinity in calculus

Infinity is a concept in calculus that helps us understand the behavior of functions as they grow larger or smaller without bound. It is not a number but a direction of sorts that indicates unbounded behavior. In the context of limits, it helps describe the behavior of a function as the input grows exceptionally large, like approaching infinity.

• "Approaching infinity" means growing larger towards an unimaginably large number without ever reaching a definitive end.
• Functions can also approach negative infinity, representing values decreasing without bound.
• Infinity is utilized in limits to analyze how functions behave under extreme conditions.

In the given exercise, \( x \to \infty \) describes the scenario where \( x \) becomes exceedingly large. In such cases, terms like \( \frac{1}{x} \) tend towards zero, as they represent fractions with ever-increasing denominators. This simplifying behavior often provides insights into the limits of functions as they evolve with massive inputs.

Evaluating limits

Evaluating limits is a fundamental technique in calculus used to determine the value a function approaches as the input approaches some value, which can be finite or infinite. The limit tells us about the trend of a function's output as inputs change and are crucial for defining derivatives and integrals.
To evaluate the limit of a function, consider these steps:

• Identify the value the input approaches. This is often denoted as \( x \to a \) or \( x \to \infty \).
• Substitute the limiting value into the function, if possible, to see how it behaves.
• Use algebraic manipulation, substitutions, or known limits to simplify the expression.

In the exercise, we had \( \lim _{x \to \infty} \cos \frac{1}{x} \). The approach involved letting \( \frac{1}{x} \to 0 \) as \( x \to \infty \). Understanding the foundational behavior of the cosine function allowed us to determine that the function trend is \( \cos(0) = 1 \). Thus, evaluating limits becomes a matter of understanding the interaction and behavior of functions within specific scenarios, often leading to insights into their asymptotic behavior or values at specific intrusions.