Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits. $$ \lim _{\theta \rightarrow \infty} \frac{\sin ^{2} \theta}{\theta^{2}-5} $$

Step-by-Step Solution

Answer

The limit is 0 as \( \theta \rightarrow \infty \).

Step 1: Initial Observation

We need to evaluate the limit of the function \( \lim_{\theta \rightarrow \infty} \frac{\sin^2 \theta}{\theta^2 - 5} \). First, notice that the sine function, \( \sin \theta \), oscillates between -1 and 1. Consequently, \( \sin^2 \theta \) oscillates between 0 and 1.

Step 2: Analyze the Numerator

Since \( \sin^2 \theta \) oscillates between 0 and 1, we can conclude that \( \sin^2 \theta \) is bounded. This means that \( \sin^2 \theta \leq 1 \) for any value of \( \theta \).

Step 3: Analyze the Denominator

As \( \theta \) approaches infinity, the term \( \theta^2 - 5 \) becomes very large since the square of \( \theta \) dominates. Specifically, \( \theta^2 - 5 \approx \theta^2 \) when \( \theta \to \infty \).

Step 4: Evaluate the Limit Behavior

Given that \( \sin^2 \theta \) is bounded (at most 1) and \( \theta^2 - 5 \) becomes very large, the fraction \( \frac{\sin^2 \theta}{\theta^2 - 5} \) approaches 0 as \( \theta \) tends to infinity. This is because dividing a bounded number by an increasingly large number tends toward 0.

Step 5: Conclusion

Based on the behavior evaluated in the previous steps, the limit evaluates to 0, because the numerator is bounded while the denominator grows to infinity.

Key Concepts

Trigonometric Functions

Trigonometric functions are foundational in calculus, particularly when dealing with periodic phenomena. The sine function, denoted as \( \sin \theta \), is one of the primary trigonometric functions. It describes a periodic wave that oscillates between -1 and 1. Consequently, when we square the sine function, as seen in \( \sin^2 \theta \), the result is always non-negative, ranging from 0 to 1.
Understanding the behavior of \( \sin \theta \) is crucial when evaluating limits and determining the so-called infinite behavior of trigonometric expressions. The important takeaway here is:

• \( \sin \theta \) oscillates between -1 and 1.
• \( \sin^2 \theta \) oscillates between 0 and 1, making it bounded.

These properties allow us to confidently say that \( \sin^2 \theta \), no matter how large \( \theta \) gets, remains within this limited range.

Infinity Limits

Understanding limits at infinity is a vital aspect of calculus, as it deals with the behavior of functions as they approach extremely large or extremely small values.
When we consider the limit \( \lim_{\theta \to \infty} \frac{\sin^2 \theta}{\theta^2 - 5} \), our primary focus is on how both the numerator \( \sin^2 \theta \) and the denominator \( \theta^2 - 5 \) behave as \( \theta \) becomes infinitely large.
To break it down:

• The numerator \( \sin^2 \theta \) is bounded between 0 and 1, as explained previously.
• The denominator \( \theta^2 - 5 \) increases, approximating \( \theta^2 \) as \( \theta \to \infty\).

The crucial insight here is that any bounded quantity over an unbounded, infinitely growing quantity results in a value tending to zero. It's akin to dividing something consistently small by something enormously large, which renders the original fraction negligible in size.

Asymptotic Behavior

In calculus, asymptotic behavior describes how a function behaves as it approaches a particular limit, often infinity. It is a useful concept for understanding the long-term behavior of functions and can provide critical insights into the tendencies of mathematical models.
Consider our function \( \frac{\sin^2 \theta}{\theta^2 - 5} \) as \( \theta \) approaches infinity. As explained, \( \sin^2 \theta \) remains constant and relatively small, while \( \theta^2 - 5 \) grows indefinitely large. The resulting limit where the fraction approaches zero reflects the characteristic asymptotic behavior:

• The function "flattens out" as it approaches a horizontal asymptote at \( y = 0 \).
• This gives a clear picture: regardless of how large \( \theta \) becomes, the effect of the growing denominator overpowers the bounded numerator.

The concept of asymptotic behavior is important because it helps us determine the long-term tendencies of functions, which would otherwise be difficult to evaluate directly.