7. Order the following functions from slowest growing to fastest growing as x→∞.
a. e^x
b. x^x
c. (ln x)^x
d. e^(x/2)

Verified step by step guidance

1. Identify the growth rates of each function as \( x \to \infty \). The functions are: \( e^x \), \( x^x \), \( (\ln x)^x \), and \( e^{x/2} \).
2. Compare the exponential functions \( e^x \) and \( e^{x/2} \). Since \( e^x = (e^{x/2})^2 \), \( e^x \) grows faster than \( e^{x/2} \).
3. Analyze the function \( (\ln x)^x \). Rewrite it using exponentials: \( (\ln x)^x = e^{x \ln(\ln x)} \). Since \( \ln(\ln x) \) grows slower than any positive power of \( x \), this growth is slower than \( e^{cx} \) for any positive constant \( c \).
4. Examine \( x^x \). Rewrite as \( x^x = e^{x \ln x} \). Since \( x \ln x \) grows faster than \( x \) or \( x \ln(\ln x) \), \( x^x \) grows faster than all the other functions.
5. Order the functions from slowest to fastest growth as \( x \to \infty \): \( e^{x/2} \), \( (\ln x)^x \), \( e^x \), and \( x^x \).

Key Concepts

Growth Rates of Functions

Understanding how functions behave as x approaches infinity is essential. Growth rates compare how quickly functions increase, with some growing polynomially, exponentially, or even faster. Recognizing these differences helps in ordering functions from slowest to fastest growth.

Exponential and Logarithmic Functions

Exponential functions like e^x grow faster than any polynomial, while logarithmic functions grow very slowly. When combined, such as (ln x)^x, the growth rate depends on the interplay between the logarithm and the exponent, requiring careful analysis.

Comparing Functions Using Limits and Logarithms

To compare growth rates rigorously, taking limits of ratios or using logarithms can simplify expressions. For example, applying logarithms transforms products and powers into sums and products, making it easier to analyze and compare the dominant terms as x→∞.