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.

Step 1: Understanding the Problem

We need to find the limit of the expression \( \frac{\sin^{2} \theta}{\theta^{2} - 5} \) as \( \theta \) approaches infinity. This involves evaluating the behavior of both the numerator and the denominator as \( \theta \) becomes very large.

Step 2: Analyze the Numerator

The numerator, \( \sin^{2} \theta \), oscillates between 0 and 1 for all values of \( \theta \). Thus, it remains bounded: \( 0 \leq \sin^{2} \theta \leq 1 \).

Step 3: Analyze the Denominator

The denominator, \( \theta^{2} - 5 \), becomes very large as \( \theta \) approaches infinity, since \( \theta^{2} \) will dominate over the constant \(-5\).

Step 4: Apply Limit Properties

Since \( \sin^{2} \theta \) is bounded and the denominator grows without bound as \( \theta \rightarrow \infty \), the entire fraction \( \frac{\sin^{2} \theta}{\theta^{2} - 5} \) tends to 0. This is because a bounded numerator divided by a denominator that increases indefinitely results in the fraction approaching 0.

Step 5: Conclusion

Therefore, the limit of the given expression as \( \theta \rightarrow \infty \) is 0.

Key Concepts

Understanding Limits

Limits are a fundamental concept in calculus. They help us describe the behavior of a function as its input approaches a particular value, which may even be infinity.
In our exercise, we are tasked with finding the limit of an expression as the variable approaches infinity.
A limit can describe:

• How a function behaves at a point
• Approach toward positive or negative infinity

Understanding how each part of a function behaves can help you determine the limit. For instance, in our problem, the numerator is bounded while the denominator increases indefinitely. This helps make the limit calculation straightforward.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are all about angles and cycles.
They repeat at regular intervals (known as periodic functions), creating waves with maximum and minimum values. In the given exercise, the function \[ ext{sin}^2 \theta\] is such that it will always give a result in the range between 0 and 1 regardless of the value of \(\theta\).
Key properties of trigonometric functions:

• Periodic behavior: they repeat their values at regular intervals.
• Bounded values: for example, \( ext{sin}^2 \theta\) is always between 0 and 1.

These properties are crucial when evaluating limits involving trigonometric functions, especially as variables tend toward infinity.

The Infinity Concept in Limits

Infinity is not a number but rather a concept that describes unbounded growth or distance.
In calculus, we use infinity to express the behavior of functions and limits that grow larger without bound.When considering limits where \(\theta\) approaches infinity, we analyze each component of an expression to understand its growth.
In the problem:

• The denominator \(\theta^2 - 5\) grows extremely large as \(\theta\) becomes very large because \(\theta^2\) dominates any constant like -5.
• The result is that the fraction's value heads towards zero due to the increasing size of the denominator.

This approach allows us to see how even bounded behaviors like those from trigonometric functions lead to precise limits when combined with expressions growing to infinity.