Problem 44 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We are required to find the limit \( \lim_{x \rightarrow -\infty} \tan^{-1} x \). This means we need to evaluate how the function \( \tan^{-1} x \) behaves as \( x \) approaches negative infinity.

Step 2: Recall the Asymptotic Behavior

Recall the inverse tangent function \( \tan^{-1} x \) approaches \( -\frac{\pi}{2} \) as \( x \rightarrow -\infty \). This is due to \( \tan^{-1} x \) being an odd function with horizontal asymptotes at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \).

Step 3: Analyze the Graph

If uncertain, analyze the graph of \( \tan^{-1} x \). As \( x \) decreases, the curve gets closer to the line \( y = -\frac{\pi}{2} \) but never actually touches it. This helps confirm our asymptotic behavior.

Step 4: Conclude the Limit

Based on the asymptotic behavior and the analysis of the graph, we have \( \lim_{x \rightarrow -\infty} \tan^{-1} x = -\frac{\pi}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions, such as \( \tan^{-1} x \), are essential in calculus due to their unique properties. These functions "invert" the usual trigonometric functions, enabling us to find an angle when given a ratio. For instance, \( \tan^{-1} \) represents the angle whose tangent is a given number. Unlike their trigonometric counterparts, inverse functions typically have restricted domains to ensure they are one-to-one and thus have valid inverses.

The function \( \tan^{-1} x \) specifically has a range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \), which means the output will always be an angle between these two values. This range is crucial because it affects how the function behaves, especially nearing infinity or negative infinity. Understanding the properties and the graphs of inverse trigonometric functions helps in solving limits problems since it can depict how these functions approach certain values asymptotically, as seen in the exercise we are working with.

Knowledge of inverse trigonometric functions' behavior strengthens the overall comprehension of calculus principles, specifically in finding and evaluating limits.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as it approaches a particular point at infinity. In the case of the function \( \tan^{-1} x \), it's vital to understand how it behaves as \( x \rightarrow -\infty \). This inverse tangent function has distinctive horizontal asymptotes.

• As \( x \to -\infty \), \( \tan^{-1} x \) approaches \( -\frac{\pi}{2} \).
• As \( x \to +\infty \), \( \tan^{-1} x \) approaches \( \frac{\pi}{2} \).

This behavior arises because the tangent's range is limitless, yet its inverse is constrained between two specific angles. Since \( \tan^{-1} x \) cannot exceed \( \pm \frac{\pi}{2} \), approaching the asymptotes makes sense when \( x \) tends to infinity or negative infinity.

Recognizing and applying asymptotic behaviors allows us to deduce the limits of trigonometric functions more intuitively. It underlines the significance of familiarity with how graphs plot and are evaluated on asymptotes.

Graphical Analysis

Graphical analysis helps provide a visual understanding of the limits and asymptotic behavior of functions like \( \tan^{-1} x \). When graphed, \( \tan^{-1} x \) takes the appearance of a curve approaching the horizontal asymptotes at \( y = -\frac{\pi}{2} \) and \( y = \frac{\pi}{2} \), without ever actually crossing these lines.

• To analyze graphically, plot \( \tan^{-1} x \) to observe how the function approaches \( -\frac{\pi}{2} \) for \( x \to -\infty \).
• The curve rises from the bottom left, flattens around \((0,0)\), then gently rises toward \( \frac{\pi}{2} \) as \( x \) increases.

Using graph plotting tools or software provides deeper insights into the function's behavior near key points or limits. Thus, graphical analysis reinforces understanding gained from algebraic or theoretical studies of limits, ensuring comprehension of both visual and conceptual aspects.