Problem 40 - Calculus: Early Transcendentals

Question

Find the limit or show that it does not exist. \( \displaystyle \lim_{x \to 0^+} \tan^{-1}(\ln x) \)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the limit of the function \( \tan^{-1}(\ln x) \) as \( x \to 0^+ \). This involves the evaluation as \( x \) approaches 0 from the right.

Step 2: Analyze the Inner Function \( \ln x \)

Consider \( \ln x \) as \( x \to 0^+ \). As \( x \) gets close to 0 from the right, \( \ln x \) approaches \(-\infty\), because the natural logarithm of a number approaching zero from the positive side tends towards negative infinity.

Step 3: Evaluate \( \tan^{-1}(y) \) as \( y \to -\infty \)

The function \( \tan^{-1}(y) \) is bounded, with horizontal asymptotes at \( y = -\pi/2 \) and \( y = \pi/2 \). Specifically, as \( y \to -\infty \), \( \tan^{-1}(y) \to -\pi/2 \).

Step 4: Combine the Results

Substitute \( y = \ln x \) into \( \tan^{-1}(\ln x) \). As \( x \to 0^+ \), \( \ln x \to -\infty \), and therefore \( \tan^{-1}(\ln x) \to -\pi/2 \).

Step 5: State the Limit

Thus, the limit is \( \lim_{x \to 0^+} \tan^{-1}(\ln x) = -\frac{\pi}{2} \).

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions allow us to determine angles when the value of a trigonometric function is known. They are the inverse operations of the trigonometric functions like sine, cosine, and tangent. A key aspect of these functions is their range, a set of output values for defined within particular intervals. For example, the function \( \tan^{-1}(x) \), or arctan, maps real numbers to values between \( -\pi/2 \) and \( \pi/2 \).
These functions are continuous and exhibit certain asymptotic properties that are particularly useful when evaluating limits, as seen in the original exercise. Understanding the behavior of \( \tan^{-1}(x) \) at positive and negative infinity helps predict how inversely related expressions behave when reaching extreme values in limit problems.

• For instance, \( \tan^{-1}(x) \to \pi/2 \) as \( x \to \infty \)
• \( \tan^{-1}(x) \to -\pi/2 \) as \( x \to -\infty \)

These properties are essential for the understanding of how \( \tan^{-1}(\ln x) \) behaves particularly when solving limits as \( x \to 0^+ \).

Natural Logarithm

The natural logarithm function, denoted \( \ln(x) \), is the inverse of the exponential function \( e^x \). It expresses the exponent to which the base \( e \) (approximately 2.718) must be raised to obtain a number \( x \).
The natural logarithm of a number approaching zero from the positive side is particularly special since:

• \( \ln(x) \to -\infty \) as \( x \to 0^+ \)

This is crucial in the context of the exercise where we analyze the limit of \( \tan^{-1}(\ln x) \). Here, \( \ln(x) \to -\infty \) transforms the problem to the study of how the arc tangent function behaves as the input diverges towards negative infinity. Understanding this is central to solving limit problems that involve logarithmic expressions approaching such boundaries.

Asymptotic Behavior

Asymptotic behavior describes how a function behaves as its input approaches a certain value or infinity. In calculus, this often involves examining the tendencies of a function at the extreme ends of its domain. It provides insights into function limits and potential horizontal or vertical asymptotes.
For the function \( \tan^{-1}(x) \):

• It approaches \( \pi/2 \) as \( x \to \infty \)
• It approaches \( -\pi/2 \) as \( x \to -\infty \)

These are horizontal asymptotes indicating that the function levels off and does not exceed these values. This characteristic plays a significant role in solving limits involving the composition of functions such as \( \tan^{-1}(\ln x) \). Understanding asymptotic behavior allows us to predict the direction and magnitude of functions at boundaries, which helps to conclude that \( \lim_{x \to 0^+} \tan^{-1}(\ln x) = -\pi/2 \).
By comprehending how functions behave asymptotically, one gains a deeper appreciation for solving calculus problems that examine the limits of tangents and logarithms at or approaching infinity or zero.