Problem 23 - Essential Calculus

Question

In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow \infty} \frac{\cos x}{x}\)

Step-by-Step Solution

Answer

\(\lim _{x \rightarrow \infty} \frac{\cos x}{x} = 0\

Step 1: Recognize the limit problem

The main task is to calculate the limit of the function \(\frac{\cos x}{x}\) as \(x\) tends towards infinity. This is in the form of \(\frac{f(x)}{g(x)}\) where \(f(x) = \cos x\) and \(g(x) = x\).

Step 2: Analyze limit properties and behaviors

Considering the function and the limit as \(x\) tends to infinity, recognize that the cosine function oscillates between -1 and 1 for all real values of \(x\). However, the denominator \(x\) tends towards infinity. The cosine function doesn't grow as rapidly as a linear function. Therefore, as \(x\) becomes large, the fraction becomes small.

Step 3: Calculate the limit

As the denominator grows much faster than the numerator, the limit when \(x\) tends to infinity of \(\frac{\cos x}{x}\) will tend towards zero. Thus, \(\lim _{x \rightarrow \infty} \frac{\cos x}{x} = 0\).

Key Concepts

L'Hôpital's Rule

When you're faced with a limit problem where you have an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), L'Hôpital's Rule can be a lifesaver. Essentially, the rule allows you to differentiate the numerator and the denominator separately and then take the limit of this new fraction.

• The rationale behind the rule is rooted in calculus, where it provides a way to simplify difficult limits into something more manageable.
• Remember, L'Hôpital's Rule can only be used if the limit you are analyzing starts in the form of \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).
• It requires the functions you are dealing with to be continuous and differentiable around the point you're considering.

In this exercise, we did not use L'Hôpital's Rule directly because the form was \( \frac{\text{bounded function}}{\infty} \), which naturally resolves to zero. Understanding when L'Hôpital's Rule is applicable is crucial in limit-solving strategies.

Infinity

Infinity in mathematics often represents concepts that endlessly grow or shrink, defying finite boundaries. When you see \( x \rightarrow \infty \), it's plain that you're dealing with values that soar beyond any defined number.

• Functions involving infinity generally lead to outcomes that grow infinitely, shrink infinitely, or oscillate without bounds.
• In this example, \( x \) growing towards infinity means we should carefully examine both the numerator and the denominator's dominant behavior.
• If one goes to infinity, while the other doesn't grow as fast, the outcome often simplifies significantly, like a fraction where the denominator's infinity dominates.

Here the critical observation is recognizing how \( \frac{\cos x}{x} \) behaves, with the cosine function remaining bounded between -1 and 1, while \( x \) infinitely grows, leading us to conclude the fraction shrinks towards zero.

Trigonometric Functions

Trigonometric functions like \( \cos x \) play a vital role in various branch of mathematics, originating from the study of angles and right triangles.

• The cosine function, \( \cos x \), consistently oscillates between -1 and 1, regardless of how large or small \( x \) gets.
• This bounded oscillation is due to the periodic nature of trigonometric functions, meaning they repeat their values in cycles.
• Thus, in limits approaching infinity, the bounded nature of \( \cos x \) often means it has a subdued influence on the entire expression it belongs to.

In our problem, \[ \lim _{x \rightarrow \infty} \frac{\cos x}{x} \]the bounded behavior of \( \cos x \) contrasts with the unbounded linear growth of \( x \). Hence, the trigonometric function's influence diminishes, allowing us to conclude the limit tends to zero.