Problem 17 - University Calculus: Early Transcendentals

Question

Find the limits. (If in doubt, look at the function's graph.) $$\lim _{x \rightarrow \infty} \sec ^{-1} x$$

Step-by-Step Solution

Answer

The limit is \( \frac{\pi}{2} \).

Step 1: Understand the Function

The function \( \sec^{-1} x \), also known as the inverse secant function, is defined for \(|x| \geq 1\). It gives the angle \( \theta \) whose secant is \ x \, and it has a natural range of \( [0, \pi/2) \cup (\pi/2, \pi] \).

Step 2: Convert to Cosine Inverse

Since \(\sec(\theta) = \frac{1}{\cos(\theta)}\), we know that \(\sec^{-1}(x) = \cos^{-1}\left(\frac{1}{x}\right)\). This helps us understand the behavior at infinity by analyzing \(\cos^{-1}\left(\frac{1}{x}\right)\).

Step 3: Analyze the Limit as x Approaches Infinity

Evaluate \(\lim_{x \to \infty} \cos^{-1}\left(\frac{1}{x}\right)\). As \(x\rightarrow \infty\), \(\frac{1}{x}\rightarrow 0\), causing the inverse cosine function \(\cos^{-1}\left(\frac{1}{x}\right)\rightarrow \cos^{-1}(0)\).

Step 4: Recognize the Result for Cosine

Since \(\cos^{-1}(0) = \frac{\pi}{2}\), the limit evaluates to this angle. So, \(\lim_{x \to \infty} \sec^{-1}(x) = \frac{\pi}{2}\).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are incredibly useful in calculus, especially when dealing with angles and lengths. These functions essentially reverse what their standard counterparts do. For instance, while the secant function is the reciprocal of the cosine, the inverse secant function, denoted as \( \sec^{-1} x \), allows us to find the angle \( \theta \) such that \( \sec(\theta) = x \).
\( \sec^{-1} x \) is only defined for values where \(|x| \geq 1\), due to the nature of the secant function, which becomes undefined for \(-1 < x < 1\). This inverse function has a restricted range of \([0, \pi/2) \cup (\pi/2, \pi]\), ensuring the function is well-behaved and trustworthy for computations.
Understanding inverse trigonometric functions like \( \sec^{-1} x \) is crucial when analyzing limits in calculus or solving trigonometric equations. They provide insight into how angles work within the unit circle.

Secant Function

The secant function, represented as \( \sec(\theta) \), is related closely to the cosine function, with the relationship \( \sec(\theta) = \frac{1}{\cos(\theta)} \). One of the main characteristics of the secant function is that it is undefined for values of \( \cos(\theta) = 0 \). This happens at angles like \( \pi/2 \), \( 3\pi/2 \), etc.
This reciprocal relationship means that the secant function helps us find the distance or lengths using angles, specifically in scenarios involving right triangles or circles. \( \sec(\theta) \) is also important for solving trigonometric limits, as it transforms inversely with \( \cos \).
When dealing with functions like \( \sec^{-1}(x) \), recognizing their relationship with cosine can make it easier to solve limits or graphs involving these functions.

Convergence at Infinity

Convergence at infinity is a concept in calculus that describes how a function behaves as its input grows very large or very small. When we talk about \( \lim_{x \to \infty} f(x) \), we are investigating what \( f(x) \) approaches as \( x \) grows beyond all bounds.
For the limit \( \lim_{x \to \infty} \sec^{-1} x \), we consider how the inverse secant function changes at extremely large numbers. By using the transformation \( \sec^{-1}(x) = \cos^{-1}\left(\frac{1}{x}\right) \), it becomes simpler to understand the behavior. As \( x \to \infty \), \( \frac{1}{x} \to 0 \), and thus \( \cos^{-1}(\frac{1}{x}) \to \cos^{-1}(0) \). This leads to the result that the limit evaluates to \( \frac{\pi}{2} \), since \( \cos^{-1}(0) = \frac{\pi}{2} \).
Understanding convergence at infinity helps in grasping how trigonometric and other functions stabilize or reach a specific value as inputs grow indefinitely.