Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 19-28, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{x \rightarrow \pi} \frac{1+\sin (x-3 \pi / 2)}{x-\pi} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Problem

We need to find the limit \( \lim _{x \rightarrow \pi} \frac{1+\sin (x-\frac{3\pi}{2})}{x-\pi} \). This involves approaching \( x \) close to \( \pi \) and determining the behavior of the function.

Step 2: Simplify the Function Inside the Limit

Observe the function \( \sin(x - \frac{3\pi}{2}) \). This can be rewritten using the identity: \( \sin(x - \frac{3\pi}{2}) = \cos(x) \) since the angle shift is equivalent to a quarter rotation.

Step 3: Substitute and Simplify

Replace \( \sin(x - \frac{3\pi}{2}) \) with \( \cos(x) \) in the limit expression: \[ \lim_{x \rightarrow \pi} \frac{1 + \cos(x)}{x-\pi}. \] As \( x \to \pi \), \( \cos(x) \to -1 \), making the numerator approach zero.

Step 4: Apply L'Hôpital's Rule

Since both the numerator and denominator approach zero, apply L'Hôpital's Rule: \[ \lim_{x \rightarrow \pi} \frac{d}{dx}(1 + \cos(x)) \Big/ \frac{d}{dx}(x - \pi) = \lim_{x \rightarrow \pi} \frac{-\sin(x)}{1}. \]

Step 5: Evaluate the Derivative at the Limit Point

Substitute \( x = \pi \) into the differentiated expression: \(-\sin(\pi)\). Since \( \sin(\pi) = 0 \), the limit simplifies to \( 0 \).

Step 6: Verify the Result Using a Graphing Calculator

Plot the function \( \frac{1 + \sin(x - \frac{3\pi}{2})}{x - \pi} \) near \( x = \pi \). Observe that the graph approaches zero as \( x \) approaches \( \pi \).

Key Concepts

L'Hôpital's Rule

In calculus, L'Hôpital's Rule is a powerful tool for finding limits, especially indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When you encounter a situation where both the numerator and denominator of a function approach zero or infinity as \( x \) approaches a limit point, L'Hôpital's Rule can be applied.
To use L'Hôpital's Rule:

• Differentiate the numerator and the denominator separately.
• Compute the limit of the new fraction. If the result is still indeterminate, apply the rule again.

When applying L'Hôpital's Rule to the limit \( \lim_{x \to \pi} \frac{1 + \cos(x)}{x - \pi} \), both the numerator \(1 + \cos(x)\) and denominator \(x - \pi\) tend to zero as \( x \to \pi \). Thus, we apply this rule to find the correct limit.

Trigonometric Identities

Trigonometric identities are crucial for simplifying expressions involving trigonometric functions. In this problem, we utilized the identity \( \sin(x - \frac{3\pi}{2}) = \cos(x) \), thanks to a quarter rotation equivalence.
Understanding such identities helps in expressing trigonometric functions in a form that's easier to work with. There are many identities, but here are some of the basic ones you should know:

• \( \sin(x) = \cos(\frac{\pi}{2} - x) \)
• \( \cos(x) = \sin(\frac{\pi}{2} - x) \)
• \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

Using these identities smartly can change complex expressions into simpler ones, aiding in finding limits or even solving complete calculus problems efficiently.

Graphing Calculator

A graphing calculator is an invaluable tool for visualizing functions and their behavior near specific points. When working on calculus problems involving limits, plotting the function can provide insights into what happens as \( x \) approaches a particular value.
For the limit \( \lim _{x \rightarrow \pi} \frac{1+ ext{sin}(x-\frac{3\pi}{2})}{x-\pi} \), using a graphing calculator allows you to:

• Visualize the approach of the function to zero as \( x \to \pi \).
• See where the function might have discontinuities or undefined behavior.
• Understand the general trend of the function, ensuring the calculated limit is correct.

Graphs help confirm analytical solutions, providing an alternative perspective on the problem.

Function Behavior Near a Point

Understanding function behavior near a certain point is critical in calculus, particularly for finding limits. The behavior of a function as it approaches a specific input value can determine whether a limit exists and what value it might have.
In examining the limit \( \lim_{x \to \pi} \frac{1 + \sin(x - \frac{3\pi}{2})}{x - \pi} \), observe how the function behaves around \( x = \pi \):

• The numerator approaches zero since \( \cos(x) \) becomes \(-1\) at \( \pi \), making \(1 + (-1) = 0\).
• The denominator also approaches zero, creating an indeterminate form \( \frac{0}{0} \).

By understanding this interaction, you can apply techniques like L'Hôpital's Rule to find the limit, ensuring a deeper comprehension of the function's behavior at that point.