Problem 17 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: The limit of the function \(\frac{\cos{\theta}}{\theta^2}\) as \(\theta\) approaches infinity is 0.

Step 1: Determine if the Limit is Indeterminate

First, let's determine if the given limit is indeterminate. As \(\theta\) approaches infinity, the denominator \(\theta^2\) approaches infinity. The numerator \(\cos \theta\) is a trigonometric function that oscillates between -1 and 1. The given limit is of the form \(\frac{f(\theta)}{g(\theta)}\), where \(f(\theta)=\cos\theta\) and \(g(\theta)=\theta^2\). Since \(g(\theta)\rightarrow\infty\) as \(\theta\rightarrow\infty\), the function is indeterminate of the form \(\frac{1}{\infty}\).

Step 2: Apply L'Hôpital's Rule

As the expression is indeterminate, we apply L'Hôpital's rule which states: $$\lim_{\theta\rightarrow\infty}\frac{f(\theta)}{g(\theta)}=\lim_{\theta \rightarrow \infty} \frac{f'(\theta)}{g'(\theta)}$$ if the resulting limit exists or is an indeterminate form of either \(0\cdot\infty, \frac{0}{0},\) or \(\frac{\infty}{\infty}\). Differentiate the numerator and denominator of the expression with respect to \(\theta\). Derivative of the numerator with respect to \(\theta\) is: $$\frac{d}{d\theta}(\cos \theta)= -\sin \theta$$ Derivative of the denominator with respect to \(\theta\) is: $$\frac{d}{d\theta}(\theta^2)= 2\theta$$ Now, let's rewrite the limit using the derivatives above.

Step 3: Evaluate the Limit of the New Expression

We have the new expression as: $$\lim_{\theta \rightarrow \infty} \frac{-\sin \theta}{2\theta}$$ Observe that \(-\sin \theta\) is bounded between -1 and 1. As \(\theta\) approaches infinity, the denominator \(2\theta\) approaches infinity. Hence, this limit is also of the form \(\frac{1}{\infty}\), which is equal to 0. Answer: $$\lim_{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^2} = 0$$

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a crucial method in calculus used to find limits of indeterminate forms. Imagine you're approaching infinity or zero, and your limit doesn't lead to a straightforward solution. This is when L'Hôpital's Rule becomes incredibly useful. It helps us simplify complex expressions by differentiating the numerator and the denominator of a fraction.Here's how it works:

• Identify if the limit is of form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). If so, L'Hôpital's Rule can be applied.


• Derivate both the numerator and the denominator separately.


• Recalculate the limit using the derivatives. Repeat the process if the indeterminate form persists.

In the case of the example \(\lim_{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^2}\), the original fraction turns out to be indeterminate. So, L'Hôpital's Rule came to the rescue by simplifying it into a much-easier-to-evaluate expression.

Indeterminate Forms

Indeterminate forms arise when calculating limits that seem undefined or ambiguous at first glance. In calculus, these forms usually look like fractions or expressions that lead us to confusion, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\).These are common indeterminate forms:

• \(\frac{0}{0}\)


• \(\frac{\infty}{\infty}\)


• \(\infty - \infty\)


• \(0 \cdot \infty\)


• \(0^0, \infty^0, \text{and } 1^\infty\)

Whenever such a form surfaces, it suggests the need for a more refined approach such as L'Hôpital's Rule. In the exercise provided, the function \(\lim_{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^2}\) initially seems indeterminate because the fraction is \(\frac{1}{\infty}\). This apparent undefined nature hints at new ways, like L'Hôpital's Rule, to evaluate the limit properly.

Trigonometric Limits

Trigonometric limits involve functions like sine, cosine, and tangent, which often show oscillating behavior. The key to understanding these limits is recognizing how these functions change as they approach certain angles or tend towards infinity.Consider these aspects:

• Functions such as \(\sin \theta\) and \(\cos \theta\) are bounded between -1 and 1 regardless of \(\theta\). This bounded nature aids in simplifying complex fractions when calculating limits.


• Trigonometric identities and known limits, such as \(\lim_{\theta \to 0}\frac{\sin \theta}{\theta} = 1\), are helpful tools.

For example, in \(\lim_{\theta \rightarrow \infty} \frac{\cos \theta}{\theta^2}\), knowing that \(\cos \theta\) does not grow or shrink beyond its bounds is crucial. As the denominator \(\theta^2\) increases infinitely, the fraction as a whole tends toward zero. This understanding of trigonometric behavior simplifies the problem, leading to a precise solution.