Problem 97 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the value. If not possible, state the reason. As \(x \rightarrow-1^{+},\) the value of arccos \(x \rightarrow\)

Step-by-Step Solution

Answer

As \(x \rightarrow-1^{+}\), the value of arccos \(x \rightarrow \pi\) or 180 degrees.

Step 1: Understand the Behavior of the arccos Function

The arccos function gives the angle whose cosine is the input value. The domain of the arccos function is \[-1, 1\]. Any values beyond this are undefined. This is important because since \(x \rightarrow-1^{+}\), it is approaching the lower boundary of the domain from the right.

Step 2: Define the Limit

As \(x \rightarrow-1^{+}\), x is approaching -1 from the right (values slightly larger than -1). This is still within the domain of the arccos function, and thus, arccos(x) is defined here.

Step 3: Calculating the Limit

As \(x \rightarrow-1^{+}\), the value of arccos \(x \rightarrow\) the value of arccos(-1). The arccos function yields an angle of \(\pi\) or 180 degrees when its input is -1.

Key Concepts

Limits in Calculus

In calculus, limits help us understand the behavior of functions as inputs approach a specific point. Here, we focus on the limit of the arccosine function as the input approaches a boundary within its domain. When we say \( x \rightarrow -1^{+} \), we mean that \( x \) is getting very close to -1, but always staying slightly greater. This notation indicates a one-sided limit, specifically from the right.

• Limits are essential for examining functions' behavior near edges or discontinuities.
• They help find function values that can't be directly substituted due to undefined regions.

Understanding limits are crucial for calculus as they form the foundation for concepts like continuity, derivatives, and integrals. In this exercise, knowing that \( x \) is moving towards -1 from the right ensures that it will remain within \([-1, 1]\), the acceptable domain for the arccos function.

Trigonometric Functions

Trigonometric functions, like cosine, relate angles to ratios derived from right triangles, making them foundational in mathematics. The arccosine function, symbolized as arccos, is the inverse of the cosine function.When dealing with \( \text{arccos}(x) \), you're identifying the angle \( \theta \) such that \( \cos(\theta) = x \).

• Arccosine is unique as its output ranges from \([0, \pi]\), representing angles in the first and second quadrants.
• The function has a specific input domain, \([-1, 1]\), due to the nature of cosine values.
This understanding is vital in determining the behavior of the function as \( x \rightarrow -1^{+} \).For instance, as \( x \) nears -1 but remains greater, arccos will yield angles approaching \( \pi \), because the cosine of \( \pi \) is precisely -1.

Domain in Mathematics

The domain of a function consists of all input values for which the function is defined. For trigonometric inverse functions like arccos, specifying the domain is crucial.In our example with arccosine, the domain is \([-1, 1]\). Any value outside this interval makes the function undefined because no real angle exists with a cosine outside this range.

• The domain ensures all inputs result in valid outputs, keeping functions working as intended.
• For arccos, the full input range is critical as it affects potential output values and where limits can be calculated.

The domain also informs us why \( x \rightarrow -1^{+} \) is possible. Since \( -1^{+} \) approaches within the valid range, the computation of arccos would yield a finite, meaningful angle, specifically \( \pi \). These concepts illustrate the importance of the domain in analyzing function limits and ensuring mathematical operations are well-defined.