Problem 7 - Essential Calculus Early Transcendentals

Question

\(1-38=\) Find the limit. Use l'Hospital's Rule where appropriate. If there is a more elementary method, consider using it. If l'Hospital's Rule doesn't apply, explain why. $$\lim _{\theta \rightarrow \pi / 2} \frac{1-\sin \theta}{1+\cos 2 \theta}$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{4} \).

Step 1: Identify the type of limit

First, substitute \( \theta = \frac{\pi}{2} \) into the expression to determine the type of indeterminate form. We find that \( \sin \left(\frac{\pi}{2} \right) = 1 \) and \( \cos(2 \times \frac{\pi}{2}) = \cos(\pi) = -1 \). This makes the expression \( \frac{1-1}{1+(-1)} = \frac{0}{0} \), an indeterminate form, which means we can use l'Hospital's Rule.

Step 2: Apply l'Hospital's Rule

To resolve the \( \frac{0}{0} \) form, we differentiate the numerator and the denominator separately. \[ f(\theta) = 1 - \sin \theta \] Differentiating gives: \[ f'(\theta) = -\cos \theta \] and the denominator: \[ g(\theta) = 1 + \cos(2\theta) \] Differentiating gives: \[ g'(\theta) = -2\sin(2\theta) = -4\sin(\theta)\cos(\theta) \].

Step 3: Evaluate the limit of the derivatives

Now, substitute \( \theta = \frac{\pi}{2} \) into the derivatives: \[ \lim_{\theta \rightarrow \pi/2} \frac{-\cos \theta}{-4 \sin \theta \cos \theta} = \lim_{\theta \rightarrow \pi/2} \frac{1}{4 \sin \theta} \] At \( \theta = \frac{\pi}{2} \), \( \sin \left(\frac{\pi}{2}\right) = 1 \), thus \[ \lim_{\theta \rightarrow \pi/2} \frac{1}{4 \times 1} = \frac{1}{4} \].

Step 4: Conclusion

The original limit \( \lim _{\theta \rightarrow \pi / 2} \frac{1-\sin \theta}{1+\cos 2 \theta} \) evaluates to \( \frac{1}{4} \) using l'Hospital's Rule.

Key Concepts

Indeterminate Forms

In calculus, certain limits present themselves as expressions that don't initially yield obvious results. These are called indeterminate forms. Specifically, we're often faced with the challenge of dealing with expressions like \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), and others when trying to find limits. These types are especially tricky because they do not have definitive answers without further analysis.

Indeterminate forms arise when substituting the limiting value into a function does not clearly indicate which way the function is heading. For instance, both the numerator and denominator might simultaneously approach zero, producing a \( \frac{0}{0} \) form, as seen in our given example. Such cases indicate potential complexities in evaluating the limit directly.

To resolve these, mathematicians use techniques such as l'Hospital's Rule. By focusing on the rate of change of the numerator and denominator, rather than their absolute values, indeterminate forms can often be converted into expressions with clear limits.

Limits in Calculus

Limits are a fundamental concept in calculus, laying the groundwork for derivatives and integrals. Essentially, a limit captures how a function behaves as it approaches a certain point. It's vital for understanding continuity, discontinuities, and the behavior of functions.

In our problem, we evaluate the limit of a function as \( \theta \) approaches \( \frac{\pi}{2} \). By substituting \( \theta = \frac{\pi}{2} \), we first identified an indeterminate form, \( \frac{0}{0} \). This identification is crucial because it confirms that normal substitution won't work, and a more advanced technique like l'Hospital's Rule is necessary.

Limits help to solve problems involving asymptotic behavior and can indicate the instantaneous rate of change when delving into derivatives. They require careful examination, as they often form the bridge from intuitive understanding to rigorous mathematical proofs.

Differentiation

Differentiation is a process in calculus used to determine the instantaneous rate of change or slope of a function at any point. Directly tied to the concept of limits, differentiation calculates the derivative of a function, providing insights into its behavior and trends.

In the context of l'Hospital's Rule, differentiation plays a pivotal role. By deriving both the numerator and the denominator, we simplify the original limit problem to a new limit that is often easier to evaluate.
In our example, we differentiated the numerator \( f(\theta) = 1 - \sin \theta \) resulting in \( f'(\theta) = -\cos \theta \), and the denominator \( g(\theta) = 1 + \cos(2\theta) \), resulting in \( g'(\theta) = -4\sin(\theta)\cos(\theta) \).

This differentiation led us to a new expression, \( \frac{-\cos \theta}{-4 \sin \theta \cos \theta} \), which we further simplified to \( \frac{1}{4 \sin \theta} \). The application of differentiation ultimately provided a clear path to evaluate the limit accurately as \( \theta \) approaches \( \frac{\pi}{2} \).

Understanding differentiation well equips students to handle more complex calculus problems, and knowing when to apply it is an essential skill in mathematics.