Problem 38 - Calculus - AP Edition

Question

Find the limits. $$\lim _{x \rightarrow+\infty}[\ln x-\ln (1+x)]$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identifying the Problem

We are asked to find the limit as \(x\) approaches infinity of the expression \( \ln x - \ln(1+x) \). This involves manipulating logarithmic functions.

Step 2: Using Logarithmic Properties

Use the logarithmic property \( \ln a - \ln b = \ln \left(\frac{a}{b}\right) \) to combine the logarithms: \[ \ln x - \ln (1+x) = \ln \left(\frac{x}{1+x}\right). \]

Step 3: Analyzing the Expression

The expression inside the logarithm is \( \frac{x}{1+x} \). As \(x\) approaches infinity, both the numerator and the denominator approach infinity.

Step 4: Simplifying the Expression

Divide the numerator and the denominator by \(x\): \[ \frac{x}{1+x} = \frac{x/x}{1/x + x/x} = \frac{1}{1/x + 1}. \]

Step 5: Finding the Limit

As \(x\) approaches infinity, \(1/x\) approaches zero. Therefore, \[ \frac{1}{1/x + 1} \to \frac{1}{0 + 1} = 1. \]

Step 6: Evaluating the Logarithm

Thus, the expression simplifies to finding \(\ln(1)\). Since \(\ln(1) = 0\), the limit is \[ \lim_{x \to +finity} \ln \left(\frac{x}{1+x}\right) = 0. \]

Key Concepts

Limits

Limits play a crucial role in calculus, providing insights into the behavior of functions as they approach specific points or infinity. In essence, a limit describes the value that a function approaches as the input (or variable) gets closer to a particular point. Understanding limits is essential for analyzing the behavior of functions at boundary points, which often leads to discovering more about their continuity and differentiability.

For example, when examining the problem provided, we aimed to find the limit of a logarithmic function as the variable approaches infinity. This typically involves transforming or simplifying the expression to make it easier to evaluate. By using the properties of logarithms, the often intricate and seemingly complex behavior of functions can reveal simple results, as demonstrated in the step-by-step solution provided.

Limits are used to calculate derivatives and integrals, which are the fundamental tools in calculus. They form the foundation for understanding instantaneous rates of change and the accumulation of quantities. Therefore, grasping the concept of limits is fundamental to not only calculus but also in fields that rely on mathematical analysis.

Logarithmic Functions

Logarithmic functions are a significant part of calculus, providing a way to deal with exponential rates of growth and decay. A logarithmic function is the inverse of an exponential function. It is widely used in various fields like biology, physics, and finance to model growth processes.

Essentially, the natural logarithm \(\ln(x)\) is a powerful tool in calculus calculations. The properties of logarithms, such as \(\ln(a) - \ln(b) = \ln\left( \frac{a}{b} \right)\), allow us to simplify complex expressions, like the one in the given problem. This property can change subtraction into a single logarithm of a fraction, facilitating easier computation by reducing the complexity of expressions.

Moreover, logarithms help bridge connections to other mathematical constructs in calculus. For instance, solving limits involving logarithms often requires a sound understanding of logarithmic properties and their behavior as variables approach infinity. The exploration of these properties aids in simplifying the analysis and solving of sophisticated problems.

Infinite Limits

Infinite limits involve evaluating what happens to a function as the variable approaches infinity or negative infinity. They help identify the end behavior of a function and show how it behaves in extreme conditions.

In the scenario of the problem from the exercise, evaluating the limit as \(x\) approaches positive infinity required understanding the behavior of both the expression and the function as a whole at extreme values. We looked at the ratio \(\frac{x}{1+x}\), simplifying it further to understand how it behaves for large values of \(x\).

The technique of dividing terms by the highest power in the denominator aids in uncovering the simplified limit as carefully demonstrated. When \(x\) becomes very large, terms that diminish to zero (like \(1/x\)) provide significant reduction of complexity and lead us to conclude the limit of the function effectively.

• Infinite limits can indicate horizontal asymptotes when relevant, providing graphical insights into function behavior.
• Solving limits at infinity helps establish where a function "settles" and is vital in sketching accurate graphs.
• Mastering infinite limits strengthens analytical skills in predicting how functions grow, shrink, and stabilize at boundaries.