Problem 17 - University Calculus: Early Transcendentals

Question

Use I'Hópital's rule to find the limits. $$\lim _{\theta \rightarrow \pi / 2} \frac{2 \theta-\pi}{\cos (2 \pi-\theta)}$$

Step-by-Step Solution

Answer

The limit is \(-2\).

Step 1: Identify the Indeterminate Form

The limit given is \( \lim_{\theta \rightarrow \pi/2} \frac{2\theta - \pi}{\cos(2\pi - \theta)} \). Let's evaluate the expression as \( \theta \to \pi/2 \). At \( \theta = \pi/2 \), the numerator becomes zero: \( 2(\pi/2) - \pi = 0 \). The denominator, \( \cos(2\pi - \pi/2) \), also becomes zero. Therefore, we have the indeterminate form \( \frac{0}{0} \).

Step 2: Apply L'Hôpital's Rule

Since the expression is in the \(\frac{0}{0}\) form, we apply L'Hôpital's Rule, which involves differentiating the numerator and denominator separately. Differentiating the numerator \(2\theta - \pi\) gives \(2\). Differentiating the denominator \(\cos(2\pi - \theta)\) involves the chain rule, resulting in \(\sin(2\pi - \theta)(-1) = -\sin(2\pi - \theta)\).

Step 3: Evaluate the Limit After Differentiation

Now, we evaluate the new limit: \( \lim_{\theta \rightarrow \pi/2} \frac{2}{-\sin(2\pi - \theta)} \). At \( \theta = \pi/2 \), the term \( \sin(2\pi - \theta) = \sin(\pi/2) = 1 \). Therefore, the limit simplifies to \(-2\).

Key Concepts

Indeterminate Forms

When calculating limits, you might find some expressions where traditional methods do not immediately provide an answer. Such cases are known as indeterminate forms. A common example is \( \frac{0}{0} \). This arises when both the numerator and denominator of a fraction tend to zero as a variable approaches a certain value. In our exercise, as \( \theta \to \pi/2 \), both the numerator \( 2\theta - \pi \) and the denominator \( \cos(2\pi - \theta) \) approach zero. Indeterminate forms like

• \( \frac{0}{0} \)
• \( \frac{\infty}{\infty} \)

require special techniques, such as L'Hôpital's Rule, to evaluate. Understanding which expressions create these forms is crucial for solving limits effectively.

Differentiation

Differentiation is the process of finding the derivative of a function. Derivatives measure how a function's output value changes as its input changes. In applying L'Hôpital's Rule, we need to differentiate both the numerator and the denominator of a given function. For the expression in the exercise, we have to differentiate:

• Numerator: \( 2\theta - \pi \)
• Denominator: \( \cos(2\pi - \theta) \)

Differentiating \( 2\theta - \pi \) with respect to \( \theta \) is straightforward, giving us \( 2 \). Differentiating \( \cos(2\pi - \theta) \) requires the chain rule. The chain rule is used because of the composition within the cosine function, resulting in \(-\sin(2\pi - \theta)\). Understanding derivatives and related rules, such as the chain rule, is essential for applying L’Hôpital’s Rule properly.

Limits in Calculus

Limits are foundational in calculus, providing the basis for defining derivatives and integrals. They describe the behavior of functions as inputs approach a certain value. For instance, to solve the limit \( \lim_{\theta \to \pi/2} \frac{2}{-\sin(2\pi - \theta)} \), we substitute \( \theta = \pi/2 \) into the expression. This gives \(-\sin(\pi/2) = -1 \), and the limit thus reduces to \(-2\). Limits can use various methods for evaluation. Techniques like substitution, factoring, and L'Hôpital's Rule are often employed to find definitive values for limits with indeterminate forms. Grasping the concept of limits helps in understanding behavior and continuity of functions in calculus. Each method provides a tool for navigating different types of limit problems.