Problem 41 - Calculus: Early Transcendentals

Question

Find the limit or show that it does not exist. \( \displaystyle \lim_{x \to \infty} \bigl[\ln (1 + x^2) - \ln (1 + x) \bigr] \)

Step-by-Step Solution

Answer

The limit is infinity.

Step 1: Recognize the Form of the Limit

We are given the expression \( \ln(1 + x^2) - \ln(1 + x) \). This can be simplified using the properties of logarithms. Recall that \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). Let's apply this property to simplify the expression.

Step 2: Simplify the Expression

Using the logarithm property \( \ln(1 + x^2) - \ln(1 + x) = \ln\left(\frac{1 + x^2}{1 + x}\right) \). Now, let's focus on the expression inside the logarithm to simplify further.

Step 3: Simplify the Fraction Inside the Logarithm

The expression \( \frac{1 + x^2}{1 + x} \) can be simplified. Divide the numerator and the denominator by \( x^2 \) to analyze the behavior as \( x \to \infty \): \[ \frac{1 + x^2}{1 + x} = \frac{\frac{1}{x^2} + 1}{\frac{1}{x^2} + \frac{1}{x}}. \] As \( x \to \infty \), both \( \frac{1}{x^2} \) and \( \frac{1}{x} \) go to 0.

Step 4: Evaluate the Limit of the Simplified Expression

Based on the simplification \( \frac{\frac{1}{x^2} + 1}{\frac{1}{x^2} + \frac{1}{x}} \rightarrow \frac{1}{0} \) as \( x \to \infty \). This simplifies to just \( x \), since the denominator \( 1/x \) goes to zero faster than \( 1/x^2 \). Therefore, \( \ln\left(\frac{1 + x^2}{1 + x}\right) \rightarrow \ln(x) \).

Step 5: Determine the Behavior of \(\ln(x)\) as \(x \to \infty\)

As \( x \to \infty \), \( \ln(x) \to \infty \). Therefore, \( \lim_{x \to \infty} \ln(x) \) is \( \infty \). Consequently, the original limit \( \lim_{x \to \infty} \bigl[\ln (1 + x^2) - \ln (1 + x)\bigr] \) also tends to \( \infty \).

Key Concepts

Logarithmic Simplification

In calculus, simplifying logarithmic expressions can make it easier to evaluate limits. One useful property of logarithms is that the difference between two logarithms can be expressed as a single logarithm of a quotient. This can be mathematically represented as:

• \( \ln(a) - \ln(b) = \ln \left( \frac{a}{b} \right) \)

This property allows you to transform the given limit expression into a more manageable form.

Applying this to the original exercise, we have

• \( \ln(1 + x^2) - \ln(1 + x) = \ln \left( \frac{1 + x^2}{1 + x} \right) \)

By doing this, we effectively reduce a subtraction into a single logarithm, allowing us to focus on simplifying the fraction inside the logarithm. This step is crucial for evaluating limits algebraically, especially as variables approach infinity.

Limit at Infinity

The concept of a limit at infinity is fundamental in understanding the behavior of functions as they grow larger and larger. When evaluating limits at infinity, you're essentially observing how the function behaves as the input becomes extremely large.

In the given problem, the expression \( \frac{1 + x^2}{1 + x} \) is simplified to determine its behavior as \( x \) tends to infinity:

• We start by simplifying: \( \frac{1 + x^2}{1 + x} = \frac{\frac{1}{x^2} + 1}{\frac{1}{x^2} + \frac{1}{x}} \).
• Because both \( \frac{1}{x^2} \) and \( \frac{1}{x} \) tend to zero as \( x \) increases, the fraction effectively simplifies to \( x \).
• This result tells us that the dominant term in the fraction is the \( x \), impacting how the logarithm behaves.

Understanding limits at infinity often involves recognizing which terms in a fraction grow slower or faster than others, and focusing on those that influence the limit's value.

Properties of Logarithms

Logarithms have specific properties that are incredibly useful in calculus. They allow for straightforward manipulation of complex expressions. Some key logarithmic properties include:

• The product rule: \( \ln(a \cdot b) = \ln(a) + \ln(b) \)
• The quotient rule: \( \ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b) \)
• The power rule: \( \ln(a^b) = b \cdot \ln(a) \)

These properties are valuable tools for simplifying and solving logarithmic equations, especially when evaluating limits.

In the original exercise, we applied the quotient rule to transform a subtraction of logarithms into a single logarithm of a quotient. This is essential for analyzing the fractional expression within the logarithm, allowing better focus on its limit behavior.

This step-by-step understanding of logarithmic properties simplifies expressions and reveals behaviors critical for finding limits.

Behavior of Logarithmic Functions

Logarithmic functions exhibit unique characteristics as their input values grow infinitely large. Understanding these characteristics helps deduce the approaches needed in solving limit problems involving logarithms.

• Logarithms increase without bounds as their input becomes very large; this is seen as they approach infinity.
• At larger values of \( x \), \( \ln(x) \) grows slower than any power of \( x \). This slow growth, however, continues indefinitely.
• By observing these patterns, you can establish that \( \ln(x) \to \infty \) as \( x \to \infty \).

In the exercise, the limit \( \lim_{x \to \infty} \ln(x) \) was considered, deriving the conclusion that it trends towards infinity.

Recognizing this behavior is essential for students to anticipate the result of the limit, enhancing their conceptual grasp of logarithmic scales and their implications in calculus.