Problem 42 - Thomas Calculus

Question

Find the limits in Exercises \(41-48 .\) (If in doubt, look at the function's graph.) $$ \lim _{x \rightarrow-1^{+}} \cos ^{-1} x $$

Step-by-Step Solution

Answer

The limit is \( \pi \).

Step 1: Understand the Problem

We are tasked with finding the limit as \( x \) approaches \(-1\) from the right (denoted by \(-1^{+}\)) of the function \( \cos^{-1}(x) \). The function \( \cos^{-1}(x) \) represents the inverse cosine, which gives us the angle whose cosine is \( x \). The domain of \( \cos^{-1}(x) \) is \([-1, 1]\).

Step 2: Define the Range of \( \cos^{-1}(x) \)

The range of the inverse cosine function \( \cos^{-1}(x) \) is \([0, \pi]\), which means it outputs values within that interval. As \( x \) approaches \(-1\) from the right, we need to determine what angle the function produces.

Step 3: Determine Behavior as \( x \to -1^+ \)

Since \( x \) is approaching \(-1\) from values greater than \(-1\), the closest value that \( x \) can take without being exactly \(-1\) is a number such as \(-0.999...\). At this value, \( \cos^{-1}(x) \) is approaching the inverse cosine of \(-1\).

Step 4: Evaluate \( \cos^{-1}(-1) \)

The value of \( \cos^{-1}(-1) \) is \( \pi \), because the cosine of \( \pi \) is \(-1\).

Step 5: Find the Limit

Since \( \cos^{-1}(x) \) is continuous and the domain includes \(-1\), the limit as \( x \to -1^+ \) of \( \cos^{-1}(x) \) is equal to \( \cos^{-1}(-1) = \pi \). Therefore, the limit is \( \pi \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the roles of the original trigonometric functions. Specifically, they allow us to find the angle when the trigonometric value, such as sine or cosine, is known.
Let's focus on the inverse cosine function, often represented as \( \cos^{-1}(x) \). This function helps us find the angle whose cosine value is \( x \).
**Key Points about \( \cos^{-1}(x) \):**

• **Domain**: The function can only accept input values in the interval \([-1, 1]\). This means only numbers from -1 to 1 can be used with the inverse cosine.
• **Range**: The output, or the angle, will be within \([0, \pi]\). This indicates that the angle solution is always between 0 and \(\pi\) radians.
• **Behavior**: As \( x \) gets closer to -1 from the right, as in this exercise, \( \cos^{-1}(x) \) moves towards the angle \( \pi \) because the cosine of \( \pi \) is exactly -1.

Understanding these properties allows you to predict the behavior of the inverse cosine function as the input approaches specific boundaries within its domain.

Continuity

When we examine the concept of continuity in functions, we are determining whether the function is unbroken and consistent over its domain. A function is continuous at a point if the limit of the function as the input approaches that point is equal to the function's value at that point.
For the inverse cosine function \( \cos^{-1}(x) \):

• **Consistency**: The function is continuous across its entire domain, \([-1, 1]\). This implies that there are no jumps or breaks in the graph within this interval.
• **At the Endpoint**: Specifically, at the endpoints -1 and 1, \( \cos^{-1}(x) \) still remains continuous. This is vital for calculating limits approaching these boundary points.
• **In Our Exercise**: As \( x \) approaches -1 from the right, the smooth continuity of \( \cos^{-1}(x) \) ensures that the limit is simply the function value at -1.

The attribute of continuity is what guarantees us that the limit as \( x \) tends to -1 from the right is indeed \( \pi \), which is the value of \( \cos^{-1}(-1) \).

Limit Approaching a Point

Finding a limit involves analyzing the behavior of a function as the variable approaches a particular point. When we look at the limit of \( \cos^{-1}(x) \) as \( x \to -1^+ \), it means we are examining what happens to \( \cos^{-1}(x) \) as \( x \) gets infinitely close to -1 from values greater than -1.
**Steps to Understand the Limit Process**:

• **Approaching from the Right**: The notation \( -1^+ \) indicates that \( x \) is very slightly greater than -1 and moving closer. This helps us focus our analysis on values like -0.9, -0.99, etc.
• **Function Evaluation**: We calculate the inverse cosine for these values, which means we are heading towards the angle that provides a cosine of these numbers close to -1.
• **Outcome**: Since \( x \to -1^+ \) leads \( \cos^{-1}(x) \) towards \( \pi \), this consistent approach shows us that the limit is simply \( \pi \).

This method of finding limits is essential, especially when dealing with endpoints and boundary values where continuity and defined function behavior allows for precise evaluation.