Problem 15 - Thomas Calculus in SI Units

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 Inverse Tangent Function

The function \( \tan^{-1}(x) \), also known as \( \arctan(x) \), is the inverse of the tangent function. It maps real numbers to an interval from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \).

Step 2: Analyzing Behavior at Infinity

As \( x \rightarrow \infty \), the slope of the tangent curve gets closer to zero, which means the inverse tangent \( \tan^{-1}(x) \) approaches its horizontal asymptote, \( \frac{\pi}{2} \). This occurs as the input \( x \) becomes very large.

Step 3: Conclude the Limit

Given that \( \tan^{-1}(x) \) approaches \( \frac{\pi}{2} \) as \( x \rightarrow \infty \), the limit of \( \tan^{-1}(x) \) is \( \frac{\pi}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to find the angle measurement when certain trigonometric values are known. Specifically, these functions are the inverse operations of the trigonometric functions like sine, cosine, and tangent. The function \( \tan^{-1}(x) \), or \( \arctan(x) \), is used to determine the angle whose tangent is \( x \). It maps inputs from the real line to the interval \(-\frac{\pi}{2}\) to \(\frac{\pi}{2}\). This means, for every real number \( x \), \( \arctan(x) \) will yield a result between these two values.

• When \( x = 0 \), \( \arctan(x) = 0 \).
• As \( x \) becomes very large, \( \arctan(x) \) approaches \( \frac{\pi}{2} \).
• If \( x \) becomes very negative, \( \arctan(x) \) approaches \(-\frac{\pi}{2} \).

Behavior at Infinity

Understanding how functions behave as their input values move towards infinity is crucial in calculus. For the function \( \tan^{-1}(x) \), as \( x \rightarrow \infty \), we are interested in what happens to the output, \( \arctan(x) \).
The tangent of an angle grows larger as the angle approaches \( \frac{\pi}{2} \). Thus, the inverse tangent returns values that edge closer to \( \frac{\pi}{2} \) as \( x \) grows. This is the behavior at infinity:

• The function becomes nearly horizontal; the rate of increase slows.
• \( \arctan(x) \) does not exceed \( \frac{\pi}{2} \) but it tends to this value.

This behavior is captured in the limit notation: \( \lim_{x \rightarrow \infty} \tan^{-1}(x) = \frac{\pi}{2} \).

Horizontal Asymptotes

Horizontal asymptotes are horizontal lines that the graph of a function approaches as \( x \) tends towards infinity or negative infinity. They are particularly important when determining the long-term behavior of a function.
For \( \tan^{-1}(x) \), the horizontal asymptote is the line \( y = \frac{\pi}{2} \):

• This indicates that no matter how large \( x \) becomes, \( \arctan(x) \) will not go beyond \( \frac{\pi}{2} \).
• The graph of \( \arctan(x) \) gently flattens out as it approaches this asymptote, illustrating the diminishing slope of \( \tan^{-1}(x) \) at large \( x \).
• Similarly, as \( x \rightarrow -\infty \), \( \arctan(x) \) approaches another horizontal asymptote at \( y = -\frac{\pi}{2} \).

These asymptotes are a graphical representation of the bounds of \( \arctan(x) \), showing how the function behaves indefinitely.