Problem 56 - Thomas Calculus

Question

Find the limits in Exercises \(51-66\) $$ \lim _{x \rightarrow \infty} x^{1 / \ln x} $$

Step-by-Step Solution

Answer

The limit is \( e \).

Step 1: Recognize the Form

The limit is presented in the form \( x^{f(x)} \) as \( x \to \infty \). To simplify this, we can assume the expression as \( y = x^{1 / \ln x} \) and then take the natural logarithm of both sides: \( \ln y = \ln(x^{1/\ln x}) \).

Step 2: Simplify the Logarithmic Expression

By using the property of logarithms, \( \ln(a^b) = b \ln a \), we can rewrite \( \ln y = \frac{1}{\ln x} \cdot \ln x \). This simplifies to: \( \ln y = 1 \).

Step 3: Exponentiate to Solve for \( y \)

By exponentiating both sides of \( \ln y = 1 \), we find \( y = e^1 = e \). Hence, the original expression simplifies, such that \( \lim_{x \to \infty} x^{1/\ln x} = e \).

Key Concepts

Exponential Functions

Exponential functions often make regular appearances in calculus problems, especially when working with limits. Here, the expression \( x^{1/\ln x} \) illustrates this notion. When we consider exponential functions in terms of limits, we look at how they behave as the variable approaches a certain value or infinity.
A standard form of exponential expression is \( a^b \), where \( a \) is the base and \( b \) is the exponent. In the given exercise, we have the base as \( x \) and the exponent as \( 1/\ln x \). Understanding how to manipulate such exponents with rules like logarithmic properties helps in solving these types of problems.

• Exponential growth often depicts increasing quantities such as populations or compound interest.
• The base \( e \) (natural exponential function) is frequently involved in growth models due to its unique properties when teamed with derivatives and integrals.

Recognizing and transforming the expression with logarithms allows us to simplify complex exponential limits, as we see in the exercise.

Logarithmic Properties

Logarithmic properties are essential tools for managing and simplifying complex mathematical expressions. In the exercise, the logarithm serves as a key to unlocking the expression \( x^{1/\ln x} \). By leveraging the property \( \ln(a^b) = b \ln a \), we simplify the expression \( \ln(y) = \ln(x^{1/\ln x}) \) into a form that is more manageable.
The comprehension of logarithmic properties isn't limited to this transformation only. They are widely applicable in solving equations, whether they be exponential, polynomial, or complex in nature. Some standard properties include:

• \( \ln(1) = 0 \)
• \( \ln(\frac{a}{b}) = \ln a - \ln b \)
• \( \ln(ab) = \ln a + \ln b \)

Understanding and applying these properties can serve as a bridge between various forms of expressions and allow for easier computation of limits.

Infinity Limits

Limits involving infinity often explore how functions behave as values move toward very large positive numbers or very small negative numbers. This concept comes into play in the problem \( \lim _{x \rightarrow \infty} x^{1 / \ln x} \). To solve the limit at infinity, we often use simplification methods like finding asymptotic behavior or applying known properties of exponential and logarithmic functions.
In general, calculating these limits revolves around answering a crucial question: What happens to the function as \( x \) escalates towards infinity? For \( x^{1/\ln x} \), the function remarkably approaches the constant \( e \).

• The numerator, in this case, remains constant as 1, whereas the denominator \( \ln x \) grows without bound.
• The crux of determining limits at infinity involves finding whether the function approaches a constant value, rises, or falls without bound.

Understanding these fundamentals aids in evaluating a wide array of functions' behavior as they extend towards infinity.