Problem 16 - University Calculus: Early Transcendentals

Question

Find the limits. (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 Arctangent Function

The function given is the arctangent, \( \tan^{-1} x \), which is the inverse of the tangent function. This function takes any real number input and outputs an angle. The possible output range of \( \tan^{-1} x \) is \( (-\pi/2, \pi/2) \).

Step 2: Analyze the Behavior as x Approaches -Infinity

As \( x \rightarrow -\infty \), we want to understand what happens to \( \tan^{-1} x \). For very large negative values of \( x \), the arctangent function approaches its lower bound, which is \( -\pi/2 \).

Step 3: Determine the Limit

Since the asymptotic behavior of \( \tan^{-1} x \) as \( x \rightarrow -\infty \) heads towards \( -\pi/2 \), we conclude that the limit of \( \tan^{-1} x \) as \( x \) approaches negative infinity is \( -\frac{\pi}{2} \). The graph of the function confirms this limit as it also shows the curve stabilizing near \( -\pi/2 \) for large negative \( x \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are special kinds of functions that allow us to find the angle corresponding to a given trigonometric ratio. They reverse the process of a standard trigonometric function, giving us an angle when provided with a sine, cosine, or tangent value. For example, the function \( \tan^{-1} x \), which is also known as the arctangent, gives us the angle whose tangent is \( x \). This function works for all real numbers.
The output range of \( \tan^{-1} x \) is slightly limited compared to tangent. Instead of producing an infinite range, \( \tan^{-1} x \) provides angles between \(-\pi/2\) and \(\pi/2\). This limitation means that no matter how large or small \( x \) gets, the output angle is constrained within these bounds.
Inverse trig functions are essential in many fields like geometry and calculus because they allow us to talk about angles and rotations directly, using simple numerical values.

Asymptotic Behavior

When we talk about asymptotic behavior, we’re discussing how a function behaves at the extremes of its domain. This kind of behavior is important to understand because it tells us about the trend of a function's values as the input grows very large or very small. In simpler terms, it's all about understanding the direction in which the function is headed.
For the function \( \tan^{-1} x \), as \( x \rightarrow -\infty \), the values of the function begin to settle close to its lower limit, which is \(-\pi/2\). The curve of \( \tan^{-1} x \) won't cross this line asymptotically, meaning it gets infinitely closer but never truly reaches or exceeds \(-\pi/2\).
Understanding the asymptotic tendencies of inverse trigonometric functions helps in sketching graphs and predicting limits. These graphs often look like smooth curves that flatten out at horizontal asymptotes.

Infinite Limits

Infinite limits are a concept used in calculus to describe the behavior of functions as they extend beyond typical value ranges—often toward positive or negative infinity. An infinite limit indicates that as inputs go toward extremes (either very large or very small), the outputs approach a specific real number or continue toward direction infinity.
In the context of the exercise, as \( x \rightarrow -\infty \) for \( \tan^{-1} x \), the function approaches \(-\frac{\pi}{2}\). Even though \( x \) gets infinitely negative, \( \tan^{-1} x \) will never exceed \(-\pi/2\). This is a finite limit essentially occurring at an infinite bound.
Recognizing when a function has an infinite limit helps in analyzing and predicting functional behavior across its domain. It also aids in understanding general trends, especially in mapping out graphs and understanding end behavior.