Question

Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. $$100^{x} ; x^{x}$$

Short answer

Answer: The function $x^x$ grows faster than $100^x$.

Step-by-step solution

Take natural logarithm of both functions

Since dealing with exponential functions can be complicated, it's helpful to take the natural logarithm of both functions to simplify the comparison. Let: $$f(x) = 100^{x} \Rightarrow \ln{f(x)} = x\ln{100}$$ and $$g(x) = x^{x} \Rightarrow \ln{g(x)} = x\ln{x}$$ Our goal is to find the limit of the ratio of these functions as x goes to infinity: $$\lim_{x \to \infty} \frac{f(x)}{g(x)}$$ This will help us determine their growth rates and which function grows faster.

Find the limit of the ratio of their natural logarithms

To simplify our comparison, we'll calculate the limit of the ratio of the natural logarithms of the functions instead: $$\lim_{x \to \infty} \frac{\ln{f(x)}}{\ln{g(x)}} = \lim_{x \to \infty} \frac{x\ln{100}}{x\ln{x}} = \lim_{x \to \infty} \frac{\ln{100}}{\ln{x}}$$ As x goes to infinity, the limit will be determined by the rate at which the denominator, \(\ln{x}\), grows.

Apply L'Hôpital's Rule

Since the limit is of the indeterminate form \(\frac{\infty}{\infty}\), we can apply L'Hôpital's Rule by finding the derivatives of the numerator and denominator and taking their limit as x goes to infinity: $$\lim_{x \to \infty} \frac{d}{dx} \left(\ln{100}\right) = 0$$ $$\lim_{x \to \infty} \frac{d}{dx} \left(\ln{x}\right) = \lim_{x \to \infty} \frac{1}{x}$$ Now, compute the limit of the ratio of their derivatives: $$\lim_{x \to \infty} \frac{\frac{d}{dx}(\ln{100})}{\frac{d}{dx}(\ln{x})} = \lim_{x \to \infty} \frac{0}{\frac{1}{x}} = 0$$

Interpret the result

Since the limit of the ratio of the natural logarithms of the functions is 0, we can conclude that the function \(100^x\) has a slower growth rate compared to \(x^x\). Thus, the function \(x^x\) grows faster than \(100^x\).

Key concepts

Exponential Growth

Exponential growth describes a process where the quantity increases by a consistent rate over a period of time. This concept is commonly represented by the mathematical expression \(a^x\), where \(a\) is a constant base and \(x\) is the exponent. It's important to note that the larger the base, the faster the growth if the exponent steadily increases. In our context with calculus, we're comparing two functions with exponential properties: \(100^x\) and \(x^x\). While both involve exponential growth, the latter can potentially grow faster, especially as \(x\) becomes very large.
To assist in understanding exponential growth further:

• Exponential functions can model real-world phenomena like populations or radioactive decay.
• The rule of thumb is: with the increase of \(x\), an exponential function tends towards infinity if the base \(a\) is greater than 1.
• The growth rate of an exponential function increases proportionally to its quantity, thanks to the consistent multiplier effect of the base.

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a special kind of logarithm that uses the base \(e\), where \(e\) is an irrational constant approximately equal to 2.71828. In mathematics, the natural logarithm is the inverse function of the exponential function \(e^x\), making it incredibly useful in simplifying complex exponential expressions.
Natural logarithms aid in solving problems involving exponential growth rates by transforming multiplicative relationships into additive ones. For instance, in the solution, calculating \(\ln(100^x)\) simplifies to \(x \ln(100)\) owing to logarithm properties. This makes it easier to compare the growth of functions. Key properties of natural logarithms include:

• \(\ln(ab) = \ln(a) + \ln(b)\)
• \(\ln(a^b) = b \ln(a)\)
• \(\ln(1/a) = -\ln(a)\)

These properties offer significant simplification when dealing with exponential functions, showcasing why they are integral in calculus and beyond.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful technique in calculus used to evaluate limits that present an indeterminate form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). The rule states that if you have a limit giving an indeterminate form, you can find the derivatives of the numerator and denominator separately, and then take the limit of the new fraction. This technique provides a way to open seemingly complex problems.
In the exercise, we used L'Hôpital's Rule to determine the limit of \(\frac{\ln(100)}{\ln(x)}\) as \(x\) goes to infinity. Without it, arriving at a conclusion regarding the growth rates of \(100^x\) and \(x^x\) would be difficult. Here's how L'Hôpital's Rule works in simple steps:

• Identify the indeterminate form.
• Take the derivative of the numerator and the derivative of the denominator.
• Evaluate the limit of the fraction of derivatives.

This rule is immensely helpful when dealing with limits in calculus, providing solutions to complex limit problems with relative ease.