Problem 9 - Calculus with Concepts in Calculus

Question

Find the given limit. $$ \lim _{x \rightarrow-\infty} \frac{\cos x}{\sqrt{x^{2}-1}} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the form of the limit

As \( x \to -\infty \), both the numerator \( \cos x \) and the denominator \( \sqrt{x^2 - 1} \) exhibit behavior that we need to analyze separately in order to find the limit of the function \( \frac{\cos x}{\sqrt{x^2 - 1}} \).

Step 2: Analyze the behavior of the numerator

The function \( \cos x \) is a periodic function that oscillates between -1 and 1 as \( x \to -\infty \). Thus, it doesn't settle to a particular value but remains bounded.

Step 3: Analyze the behavior of the denominator

For the denominator \( \sqrt{x^2 - 1} \), as \( x \to -\infty \), \( x^2 \) becomes very large, and \( \sqrt{x^2 - 1} \) approaches \( \sqrt{x^2} = |x| \). Since \( x \to -\infty \), \(|x| = -x\).

Step 4: Combine the results and simplify

The limit then becomes \( \frac{\cos x}{-x} \) as \( x \to -\infty \). Considering \( \cos x \) is bounded between -1 and 1, \( \frac{\cos x}{-x} \to 0 \) because the denominator \(-x\) grows unbounded.

Step 5: Conclude the limit

We conclude that because \( \cos x \) remains bounded while \(-x\) approaches -\(\infty\), the entire expression approaches 0.

Key Concepts

Behavior of Trigonometric Functions

Trigonometric functions like sine and cosine are known for their periodic and oscillating nature. Specifically, the cosine function, denoted as \( \cos x \), varies between -1 and 1. This is because \( \cos x \) represents the x-coordinate of a point on the unit circle as it moves around.

When evaluating limits involving \( \cos x \), especially as \( x \) approaches infinity or negative infinity, remember:

• \( \cos x \) does not converge to a particular value.
• Its graph forms oscillations that do not widen or settle.

Understanding this behavior is vital because it helps determine how \( \cos x \) interacts within larger expressions, influencing whether an entire function can approach a specific value.

Infinite Limits

Infinite limits explore the behavior of functions as either the input or the output becomes infinitely large or infinitely negative.

In the problem \( \lim_{x \to -\infty} \frac{\cos x}{\sqrt{x^2 - 1}} \), we're examining how the function behaves as \( x \) heads towards negative infinity.

• The denominator \( \sqrt{x^2 - 1} \) simplifies to \( -x \) as \( x \to -\infty \) because \( \sqrt{x^2} = |x| \) and for negative \( x \), \( |x| = -x \).
• The numerator \( \cos x \), however, remains oscillating between -1 and 1 regardless of \( x \)'s growth or decay.

This analysis shows how functions behave when approaching infinities, revealing the balance between numerator and denominator growths, which ultimately determines the limit's value.

Bounded Functions in Limits

Certain functions, known as bounded functions, do not exceed particular maximum or minimum values.

In our example, \( \cos x \) is a bounded function:

• Even for very large or very small \( x \), its output is confined between -1 and 1.
• This bounded nature profoundly impacts limits, especially when combined with functions that tend to infinity in the denominator, like \( -x \).

When we divide a bounded function like \( \cos x \) by a function tending to infinity like \( -x \), the entire expression shrinks towards zero. This is because the bounded numerator cannot keep up with the unbounded growth of the denominator, driving the limit to zero in such cases. Thus, understanding bounded functions helps unveil the ultimate behavior of more complex expressions in calculus limits.