Question

Show that \(x^{x}\) grows faster than \(b^{x}\) as \(x \rightarrow \infty,\) for \(b>1\)

Short answer

Question: Show that the function \(x^x\) grows faster than the function \(b^x\) as \(x\) tends to infinity, where \(b > 1\). Answer: The limit of the ratio of \(x^x\) and \(b^x\) as \(x\) approaches infinity is infinite, proving that \(x^x\) grows faster than \(b^x\) for \(b > 1\).

Step-by-step solution

Write down the ratio of the functions

Let's write down the ratio of the given functions as \(\frac{x^x}{b^x}\).

Take the natural logarithm of the ratio

We'll take the natural logarithm of the ratio to simplify the expressions and make it easier to evaluate the limit. Let \(y = \frac{x^x}{b^x}\). Taking the natural logarithm of both sides, we have: \(\ln{y} = \ln{\left(\frac{x^x}{b^x}\right)}\) Using properties of logarithms, we can simplify this expression further: \(\ln{y} = x(\ln{x} - \ln{b})\) Now, let's exponentiate both sides to get rid of the natural logarithm: \(y = e^{x(\ln{x} - \ln{b})}\)

Evaluate the limit as x approaches infinity

Now we'll evaluate the limit of this expression as \(x\) approaches infinity: \(\lim_{x \rightarrow \infty} e^{x(\ln{x} - \ln{b})}\) Since \(b > 1\), \(\ln{b} > 0\), and as \(x\) approaches infinity, \(\ln{x} > \ln{b}\). This means that the term inside the exponential expression will become infinite as \(x\) approaches infinity, resulting in an infinite limit. Therefore, the limit of the ratio of \(x^x\) and \(b^x\) as \(x\) approaches infinity is infinite, which shows that \(x^x\) grows faster than \(b^x\) for \(b > 1\).

Key concepts

Limits at Infinity

Understanding the behavior of functions as we approach very large values is crucial in mathematics. Limits at infinity refer to what value a function approaches as the variable within it grows without bound. When looking at limits as the variable, commonly named x, approaches infinity, we are essentially analyzing the end-behavior of a function. If a function approaches a certain constant value, then that constant is the limit at infinity. However, in some cases, a function might grow indefinitely, and we say its limit at infinity is either positive or negative infinity.

For the function f(x) = x^x, as x becomes extremely large, the function's value grows without bound, indicating that its limit at infinity is positive infinity. This concept is foundational in calculus and helps in understanding the long-term behavior of functions, such as in our exercise where we showed that x^x grows faster than b^x for any b greater than 1.

Exponential Growth

The term exponential growth describes a pattern in which a quantity increases by a consistent multiplicative rate at regular intervals. This type of growth is characterized by the equation f(x) = a^x, where a is the growth factor and x is the exponent or power. For a > 1, the function exhibits exponential growth, with its value quickly becoming very large as x increases.

This concept is ubiquitous in real-world scenarios such as population growth, compound interest, and certain biological processes. In our exercise, the function x^x exhibits an even more rapid increase than a regular exponential function since the base itself is growing alongside the exponent, showcasing a quintessential example of exponential growth outpacing regular exponential functions.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm with the base e, where e is the Euler's number, approximately equal to 2.71828. This function is the inverse of the exponential function e^x, meaning that if y = e^x, then x = ln(y). There are numerous properties of the natural logarithm, such as ln(1) = 0 and ln(e) = 1.

In our exercise, applying the natural logarithm simplified the ratio of x^x and b^x, which facilitated finding the limit at infinity. It's a powerful tool in calculus for disentangling complex expressions, especially when dealing with limits and growth rates.

Properties of Logarithms

Logarithms have unique properties that make them invaluable in simplifying expressions and solving equations. Some essential properties of logarithms include:

• Product Rule: ln(a * b) = ln(a) + ln(b)
• Quotient Rule: ln(a / b) = ln(a) - ln(b)
• Power Rule: ln(aⁿ) = n * ln(a)

These properties were used in the solution to the exercise when evaluating the ratio of x^x to b^x as x approaches infinity. Specifically, the logarithms allowed us to transform multiplication and exponentiation within the ratio into a simpler, more manageable algebraic form. Recognizing and properly applying these properties are vital skills in higher mathematics, particularly in calculus, enabling us to understand and analyze the behavior of complex functions.