Problem 11 - Calculus Early Transcendentals

Question

Evaluate the following limits. $$\lim _{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^{2}}$$

Step-by-Step Solution

Answer

Question: Evaluate the limit of the function \(f(\theta) = \frac{\cos \theta}{\theta^2}\) as \(\theta \rightarrow \infty\). Answer: The limit of the function as \(\theta \rightarrow \infty\) is 0.

Step 1: Rewriting the limit using Big O notation

The function in question is a rational function with a trigonometric function in the numerator that has a bounded range. To prove the behavior of this function as the input tends to infinity, we will rewrite the function using Big O notation. Denote the function as \(f(\theta) = \frac{\cos \theta}{\theta^2}\). Because \(\cos \theta\) oscillates between -1 and 1, we can rewrite the function as \(f(\theta) = \frac{O(1)}{\theta^2}\). Big O notation is used here to show that the numerator is of order one as the input grows large.

Step 2: Using a comparison limit

To further analyze the limit, we will use a comparison limit which involves a simpler function to conclude the behavior of the function \(f(\theta)\). Notice that \(0 \leq \left| \frac{\cos \theta}{\theta^2} \right| \leq \frac{1}{\theta^2}\) for all \(\theta\). As \(\theta \to \infty\), the right-hand side \(\frac{1}{\theta^2} \to 0\).

Step 3: Applying the squeeze theorem

We can use the squeeze theorem from calculus to find the limit of the function \(f(\theta)\). The squeeze theorem states that if \(g(\theta) \leq f(\theta) \leq h(\theta)\) for all \(\theta\) near some limit point, and if \(\lim_{\theta \to a} g(\theta) = \lim_{\theta \to a} h(\theta) = L\), then \(\lim_{\theta \to a} f(\theta) = L\). In our case, \(g(\theta) = 0\), \(h(\theta) = \frac{1}{\theta^2}\), and the limit point is \(\infty\). Since we have shown that \(\lim_{\theta \rightarrow \infty} \frac{1}{\theta^2} \rightarrow 0\), according to the squeeze theorem, we can conclude that:

Step 4: Evaluating the given limit

We now find the limit of the function by applying the squeeze theorem in the previous step. $$\lim _{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^{2}} = 0$$

Key Concepts

Big O Notation

Big O Notation is a way to describe the behavior of functions in terms of their growth rates. It provides a useful way to compare the efficiency of algorithms and understand how functions behave as inputs grow large. When we say that a function is \(O(n)\), we're saying the function grows at a rate proportional to \(n\), up to a constant factor, for large \(n\).
Consider our function \(f(\theta) = \frac{\cos \theta}{\theta^2}\). In this function, the numerator \(\cos \theta\) oscillates between -1 and 1, making it a constant function with bounded behavior. This bounded behavior can be denoted as \(O(1)\).
Therefore, the Big O notation helps us simplify the limit problem by indicating that the oscillation of \(\cos \theta\) is negligible as \(\theta\) grows, compared to the \(\theta^2\) in the denominator. This makes it easier to predict the function's behavior as \(\theta\) approaches infinity.

Squeeze Theorem

The Squeeze Theorem is a mathematical concept used to find limits of functions that are "squeezed" between two other functions. It's like a sandwich for limits! This theorem states that if you have three functions, \(g(\theta)\), \(f(\theta)\), and \(h(\theta)\), such that \(g(\theta) \leq f(\theta) \leq h(\theta)\) for all \(\theta\) near some limit \(a\), and if \(\lim_{\theta \to a} g(\theta) = \lim_{\theta \to a} h(\theta) = L\), then \(\lim_{\theta \to a} f(\theta) = L\) as well.
For our function \(f(\theta) = \frac{\cos \theta}{\theta^2}\), we can use the inequality:

• \(0 \leq \left| \frac{\cos \theta}{\theta^2} \right| \leq \frac{1}{\theta^2} \)

By evaluating the limits:

• The limit of \(g(\theta) = 0\) as \(\theta \to \infty\) is 0.
• The limit of \(h(\theta) = \frac{1}{\theta^2}\) as \(\theta \to \infty\) is also 0.

Therefore, by the squeeze theorem, \(\lim_{\theta \to \infty} \frac{\cos \theta}{\theta^2} = 0\). The function \(f(\theta)\) is effectively "squeezed" to 0 as well.

Trigonometric Limits

Trigonometric limits are helpful in solving problems involving trigonometric functions, especially as they approach infinity or zero.\( \cos \theta \) is a basic trigonometric function that ranges between -1 and 1 for all \(\theta\). This bounded behavior is crucial when considering limits involving \( \cos \theta \) since it simplifies our calculations by confining the function's output within a known interval.
In this particular problem, we see that the behavior of \( \cos \theta \) doesn't change as \( \theta \to \infty \). Its oscillation remains consistent between -1 and 1, and while fast or slow oscillation may not directly affect the overall behavior when combined with an asymptotically significant change like the \( \theta^2 \) in the denominator. Therefore, understanding how trigonometric functions behave is key in evaluating trigonometric limits efficiently. These limits often involve familiar identities or inequalities that simplify even seemingly complicated expressions.