Problem 32 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 31-32: Complete the statements. (a) As \(x \rightarrow 1^{-}, \sin ^{-1} x \rightarrow\) (b) As \(x \rightarrow-1^{+}, \cos ^{-1} x \rightarrow\) (c) As \(x \rightarrow-\infty, \tan ^{-1} x \rightarrow\)

Step-by-Step Solution

Answer

(a) \(\pi/2\); (b) \(\pi\); (c) \(-\pi/2\)."

Step 1: Evaluate the Limit of Inverse Sine Function

As \(x \to 1^-\), the value of \(\sin^{-1} x \) approaches the angle whose sine is approaching 1 from the negative side. The angle whose sine is 1 is \(\frac{\pi}{2}\). Thus, \(\sin^{-1} x \to \frac{\pi}{2}\) as \(x \to 1^-\).

Step 2: Evaluate the Limit of Inverse Cosine Function

As \(x \to -1^+\), the value of \(\cos^{-1} x \) approaches the angle whose cosine is approaching -1 from the positive direction. The angle whose cosine is -1 is \(\pi\). Therefore, \(\cos^{-1} x \to \pi\) as \(x \to -1^+\).

Step 3: Evaluate the Limit of Inverse Tangent Function

As \(x \to -\infty\), the value of \(\tan^{-1} x \) approaches the angle whose tangent value becomes increasingly negative. The angle for which this is true is \(-\frac{\pi}{2}\). Therefore, \(\tan^{-1} x \to -\frac{\pi}{2}\) as \(x \to -\infty\).

Key Concepts

Limits of Trigonometric Functions

Limits of trigonometric functions involve understanding how these functions behave as the variable approaches certain points or infinity. This is crucial for calculating the limits of inverse trigonometric functions.
Inverse trigonometric functions often involve angles where the original trigonometric functions are evaluated.

• Sine: As value approaches 1, the limit is \(\frac{\pi}{2}\)
• Cosine: As value approaches -1, the limit is \(\pi\)
• Tangent: As value approaches negative infinity, the limit is \(-\frac{\pi}{2}\)

The behavior of limits is fundamental for solving calculus problems involving curves, patterns, and changes in functions.

Inverse Sine

The inverse sine function, denoted as \(\sin^{-1} x\), tells us the angle whose sine is x. This function is pivotal in trigonometry and calculus, especially when solving equations involving angles and arcs.
For the range of \(\sin^{-1} x\):

• The function takes input from \(-1\) to \(1\)
• The output, which is the angle, ranges from \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\)

As the value approaches 1 from the negative direction (\(1^-\)), the angle approaches \(\frac{\pi}{2}\). This result is fundamental when considering the maximum possible value on the sine curve.

Inverse Cosine

The inverse cosine function, noted as \(\cos^{-1} x\), gives the angle whose cosine value is x. This inverse is essential in trigonometry for calculating angles from cosine values.
The range of \(\cos^{-1} x\):

• The input range is from \(-1\) to \(1\)
• The output angle ranges from 0 to \(\pi\)

As \(x\) approaches -1 from the positive side, the resulting limit is \(\pi\). This indicates that at these extreme values, the cosine angle reaches its maximum point within its primary range.

Inverse Tangent

The inverse tangent function, represented as \(\tan^{-1} x\), identifies the angle whose tangent equals x. It's useful in scenarios requiring conversion from slope to angle, for instance, in navigation and physics.
For \(\tan^{-1} x\):

• This function accepts input from negative to positive infinity
• The resulting angles lie within the interval \((-\frac{\pi}{2}, \frac{\pi}{2})\)

When \(x\) goes towards negative infinity, it results in an angle tending towards \(-\frac{\pi}{2}\). This behavior is key when analyzing real-life contexts where tangent approaches extreme negative slopes, showing high level computations in fields like engineering.