Problem 22 - University Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

The limit is \(-\frac{1}{2}\).

Step 1: Identify indeterminate form

Begin by verifying whether \( \lim _{x \rightarrow \frac{\pi}{2}} \frac{\ln (\csc x)}{(x-\frac{\pi}{2})^{2}} \) results in an indeterminate form. We consider the numerator: \( \ln(\csc x) = \ln(\frac{1}{\sin x}) = -\ln(\sin x) \). As \( x \to \frac{\pi}{2} \), \( \ln(\sin x) \) tends towards \( \ln(1) = 0 \). Hence, the numerator approaches 0. The denominator \( (x-\frac{\pi}{2})^2 \) also approaches 0. The limit is of the indeterminate form \( \frac{0}{0} \), allowing us to apply L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule

With the limit in the form \( \frac{0}{0} \), apply L'Hôpital's Rule by differentiating the numerator and the denominator separately. The derivative of the numerator \( -\ln(\sin x) \) is \( -\frac{1}{\sin x} \cdot \cos x = -\cot x \). The derivative of the denominator \( (x-\frac{\pi}{2})^2 \) is \( 2(x-\frac{\pi}{2}) \). Therefore, using L'Hôpital's Rule, the limit to evaluate becomes \( \lim _{x \rightarrow \frac{\pi}{2}} \frac{-\cot x}{2(x-\frac{\pi}{2})} \).

Step 3: Simplify and Evaluate the Limit

Substitute \( \cot x = \frac{\cos x}{\sin x} \) and consider \( x \rightarrow \frac{\pi}{2} \). As \( x \to \frac{\pi}{2} \), \( \cot x \to 0 \) and the expression depends on \( \frac{\cos x}{x-\frac{\pi}{2}} \). Using the small-angle approximation or further L'Hôpital's Rule for \( \cot x \) if necessary, evaluate:\( \frac{-\cot x}{2(x-\frac{\pi}{2})} = \frac{-\frac{1}{\tan x}}{2} \rightarrow \frac{-1}{2} \cdot \frac{1}{1} \). This simplifies to the limit \(-\frac{1}{2}\).

Key Concepts

Limit of functions

In calculus, the concept of limits forms the foundation for other advanced topics such as derivatives and integrals. The limit of a function is essentially the value that the function output approaches as the input approaches some particular point. Limits help us understand the behavior of functions near certain points, which might not necessarily be defined at those points.

To solve a problem involving limits, like in the exercise with the function \( \lim _{x \rightarrow \frac{\pi}{2}} \frac{\ln (\csc x)}{(x-\frac{\pi}{2})^{2}} \), you identify how both the numerator and denominator behave as \( x \) approaches the specified point. In our exercise, as \( x \) approaches \( \frac{\pi}{2} \), both the numerator and the denominator approach 0, creating a condition that allows us to apply L'Hôpital's Rule.

**Key Points to Remember When Calculating Limits:**

• Identify the behavior of the function as \( x \) approaches the limit point.
• Check the form of the limit, especially if it's an indeterminate form like \( \frac{0}{0} \).
• If possible, simplify the expression to make calculation easier or apply rules such as L'Hôpital's Rule if necessary.

Indeterminate forms

In calculus, indeterminate forms occur when the limit of a function cannot be directly determined because the form of the limit doesn't seem to provide clear information. These forms include \( \frac{0}{0} \), \( \frac{\infty}{\infty} \), \( 0 \cdot \infty \), among others. In our exercise, the form \( \frac{0}{0} \) appeared when evaluating the limit, which justifies the application of techniques like L'Hôpital's Rule to resolve it.

When dealing with indeterminate forms, especially \( \frac{0}{0} \), L'Hôpital's Rule proves particularly useful. This rule allows you to differentiate the numerator and the denominator separately to find the limit. However, this can only be applied if both the numerator and the denominator are continuously differentiable near the point of interest.

**Handling Indeterminate Forms Effectively:**

• First verify if the limit is in an indeterminate form.
• Use L'Hôpital's Rule as long as its conditions are fulfilled.
• Sometimes repeated applications of the rule are required if indeterminate forms persist.
• Always examine if simplifications can clarify the limit without applying advanced rules.

Calculus problem solving

Problem solving in calculus involves a systematic approach to tackling complex mathematical expressions. Let's break down these steps using the example provided in the exercise.

When you face a **calculus problem:**

• Start by identifying the problem: Understand the given function and what is being asked — here, finding the limit using L'Hôpital's Rule.
• Assess the components of the problem: Break down the function to evaluate each part's behavior as \( x \) approaches the limit. In our example, examining the behavior of \( \ln(\csc x) \) and \( (x-\frac{\pi}{2})^2 \) was crucial.
• Apply suitable theorems or rules: Based on the behavior, apply appropriate calculus rules — like the L'Hôpital's Rule — to evaluate the limit effectively.
• Simplify and calculate: Carry out the differentiation, simplification, and final calculation to solve the problem.
• Verify your answer: Double-check calculations and logic to ensure accuracy.

By using a structured approach, solving calculus problems becomes more manageable and ensures a comprehensive understanding of core concepts.