Problem 36 - Single Variable Calculus

Question

Find the limit. $$ \lim _{x \rightarrow \infty} \tan ^{-1}(\ln x) $$

Step-by-Step Solution

Answer

As \(x\) approaches infinity, \(\ln(x)\) approaches infinity, which causes the argument of the inverse tangent function to increase without bound, leading us closer to the horizontal asymptote of the inverse tangent function at \(y = \frac{\pi}{2}\). Therefore, we can conclude that: \[ \lim_{x \rightarrow \infty} \tan^{-1}(\ln x) = \frac{\pi}{2}. \]

Step 1: Recalling the properties of the inverse tangent and natural logarithm functions

The inverse tangent function (which can be written as \(\tan^{-1}(x)\) or \(\arctan(x)\)) has horizontal asymptotes at \(y = \frac{\pi}{2}\) as \(x \rightarrow \infty\) and at \(y = -\frac{\pi}{2}\) as \(x \rightarrow -\infty\). The natural logarithm function, \(\ln(x)\), is only defined for positive values of \(x\), and has the property that \(\ln(x) \rightarrow \infty\) as \(x \rightarrow \infty\).

Step 2: Analyzing the composition of the functions

Given that \(\ln(x)\) becomes infinitely large as \(x \rightarrow \infty\), we can think of the argument of the inverse tangent function getting bigger and bigger in the expression \(\tan^{-1}(\ln x)\). This will lead us closer to the horizontal asymptote of the inverse tangent function, which is at \(y = \frac{\pi}{2}\).

Step 3: Computing the limit

As \(x\) approaches infinity, \(\ln(x)\) approaches infinity too, which causes the argument of the inverse tangent function to increase without bound. Thus, \(\tan^{-1}(\ln x)\) approaches its horizontal asymptote. Therefore, we can conclude that: \[ \lim_{x \rightarrow \infty} \tan^{-1}(\ln x) = \frac{\pi}{2}. \]

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are functions that reverse the action of standard trigonometric functions. They are particularly useful when we need to retrieve angles from given trigonometric values. Here, we focus on the inverse tangent function, denoted as \( \tan^{-1}(x) \) or sometimes \( \arctan(x) \).
The inverse tangent function is defined for all real numbers and produces results between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). It reflects the angle whose tangent is the given value. A key feature of \( \tan^{-1}(x) \) is its horizontal asymptotes. These are the values the function can approach but never actually reach.

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

These asymptotes help in understanding the behaving pattern of \( \tan^{-1}(x) \), especially when dealing with limits and compositions with other functions.

Natural Logarithm

The natural logarithm, written as \( \ln(x) \), is one of the fundamental functions in mathematics. It is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental number approximately equal to 2.718. The natural logarithm has several important properties:

• It is only defined for \( x > 0 \), as the logarithm of a non-positive value is undefined.
• As \( x \) becomes very large, \( \ln(x) \) also increases without bound, meaning \( \ln(x) \to \infty \) as \( x \to \infty \).

Understanding the behavior of \( \ln(x) \) is crucial for computing limits, especially when it's part of a composition, like \( \tan^{-1}(\ln x) \). In such cases, whether \( x \) is approaching zero, a positive constant, or infinity can significantly impact the limit's result.

Horizontal Asymptotes

Horizontal asymptotes are a concept in calculus that describe a specific type of behavior of functions as the input either increases or decreases indefinitely. They indicate the value a function approaches, but does not necessarily reach, as \( x \to \pm\infty \).
For inverse trigonometric functions such as \( \tan^{-1}(x) \), these asymptotes provide valuable information:

• \( \tan^{-1}(x) \to \frac{\pi}{2} \) as \( x \to \infty \).
• \( \tan^{-1}(x) \to -\frac{\pi}{2} \) as \( x \to -\infty \).

This knowledge simplifies the evaluation of complex limits involving these functions. In the case of \( \tan^{-1}(\ln x) \), since \( \ln(x) \to \infty \) as \( x \to \infty \), the expression \( \tan^{-1}(\ln x) \) also approaches the horizontal asymptote at \( \frac{\pi}{2} \). This concept is key in solving limit problems that involve both logarithmic and inverse trigonometric functions.