Problem 53 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understand the Limit

We need to find \( \lim_{x \rightarrow \infty} (\ln x)^{1/x} \). This involves evaluating the behavior of the expression as \( x \) approaches infinity.

Step 2: Transform to Exponential Form

To simplify the expression, note that we can rewrite \( y = (\ln x)^{1/x} \) in exponential form as \( y = e^{\frac{1}{x} \ln (\ln x)} \). This transformation helps evaluate the limit.

Step 3: Evaluate the Exponent Limit

Determine the limit of the exponent: \( \lim_{x \rightarrow \infty} \frac{1}{x} \ln (\ln x) \). As \( x \rightarrow \infty \), both \( \ln x \) and \( \ln(\ln x) \) grow, but the \( \ln(\ln x) \) grows slower than \( x \), making the fraction approach zero.

Step 4: Apply Limit to Exponential Form

Since \( \lim_{x \rightarrow \infty} \frac{1}{x} \ln(\ln x) = 0 \), we substitute into the exponential form: \( \lim_{x \rightarrow \infty} e^{\frac{1}{x} \ln(\ln x)} = e^0 = 1 \). This means the original limit also approaches 1 as \( x \rightarrow \infty \).

Key Concepts

Logarithmic Functions

Logarithmic functions, abbreviated as "log," are the inverses of exponential functions. When you see an expression like \( \ln x \), this refers to the natural logarithm, which is logarithms with base \( e \), where \( e \) is approximately 2.718.
Logarithmic functions have some distinctive features:

• They grow very slowly compared to linear or polynomial functions.
• Their domain consists of positive real numbers.
• The range of a logarithmic function is all real numbers.

For example, \( \ln(1) = 0 \) and \( \ln(e) = 1 \). The exercise deals with \( \ln x \), and part of solving the limit involves understanding how it behaves as \( x \to \infty \). Given their growth behavior, logarithmic functions are crucial in problems involving limits and asymptotic behavior, especially when dealing with infinite quantities.

Exponential Growth

Exponential growth refers to a process where a quantity increases by a fixed percentage over equal time periods. In mathematics, this is often modeled using the exponential function \( e^x \), which grows faster than linear or polynomial functions.
In our problem, exponential growth is utilized to simplify the expression \( y = (\ln x)^{1/x} \) into an exponential form \( y = e^{\frac{1}{x} \ln(\ln x)} \).

• This form allows us to leverage the properties of exponentials for limit evaluation.
• When exponential functions contain terms that tend to zero, like \( e^0 = 1 \), they stabilize.

Understanding exponential growth is vital because it helps explain transforming complex expressions into manageable forms. Such transformations enable us to apply limits more effectively and evaluate behaviors at extreme values.

Asymptotic Behavior

Asymptotic behavior describes how functions behave as they approach a boundary or infinity. The term "asymptotic" often involves getting "infinitely close" to a curve but never quite reaching it. In calculus, limits help investigate these behaviors, especially when functions are complex.
In the given exercise, asymptotic behavior is key to determining \( \lim_{x \to \infty} (\ln x)^{1/x} \). Consider that as \( x \) approaches infinity, the term \( \frac{1}{x} \ln(\ln x) \) becomes exceedingly small.

• Thus, it tends to zero, which significantly impacts the entire expression's limit.
• Ultimately, the function approaches the value \( e^0 = 1 \).

The analysis of asymptotic behaviors provides insights into whether functions reach stable values or continue to diverge. This is essential in understanding the limits involving logarithmic and exponential functions.