Problem 64 - Single Variable Calculus

Question

a. Prove that \(\lim _{x \rightarrow \infty} \frac{x^{k}}{e^{x}}=0\) for every positive constant \(k\). This shows that the natural exponential function approaches infinity faster than any power function. b. Prove that \(\lim _{x \rightarrow \infty} \frac{\ln x}{x^{k}}=0\) for every positive constant \(k\). This shows that the natural logarithmic function approaches infinity slower than any power function.

Step-by-Step Solution

Answer

In summary, we proved that \(\lim _{x \rightarrow \infty} \frac{x^{k}}{e^{x}}=0\) for every positive constant \(k\) and \(\lim _{x \rightarrow \infty} \frac{\ln x}{x^{k}}=0\) for every positive constant \(k\) using L'Hôpital's rule. This shows that the natural exponential function approaches infinity faster than any power function and the natural logarithmic function approaches infinity slower than any power function.

Step 1: Use L'Hôpital's Rule

Since \(\lim_{x \rightarrow \infty} x^{k} = \infty\) and \(\lim_{x \rightarrow \infty} e^x = \infty\), we can apply L'Hôpital's rule, which states that if \(\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{\infty}{\infty}\), then \(\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \lim_{x \rightarrow c} \frac{f'(x)}{g'(x)}\), if the latter limit exists.

Step 2: Find the derivatives of the numerator and denominator

First, differentiate \(x^k\) with respect to \(x\), resulting in \(k\cdot x^{k-1}\). Then, differentiate \(e^x\) with respect to \(x\), resulting in \(e^x\).

Step 3: Apply L'Hôpital's Rule to the limit

Using L'Hôpital's Rule, we will now find \(\lim_{x \rightarrow \infty} \frac{k\cdot x^{k-1}}{e^x}\).

Step 4: Determine the new limit

Since \(\lim_{x \rightarrow \infty} k\cdot x^{k-1} = \infty\) and \(\lim_{x \rightarrow \infty} e^x = \infty\), we can continue applying L'Hôpital's Rule until the exponent on \(x\) in the numerator is \(0\). When this happens, we have \(\lim_{x \rightarrow \infty} \frac{k!}{e^x} = 0\), because the exponential function grows faster than the factorial function. Therefore, \(\lim _{x \rightarrow \infty} \frac{x^{k}}{e^{x}}=0\) for every positive constant \(k\). b. Proving \(\lim _{x \rightarrow \infty} \frac{\ln x}{x^{k}}=0\) for every positive constant \(k\)

Step 1: Use L'Hôpital's Rule

Since \(\lim_{x \rightarrow \infty} \ln x = \infty\) and \(\lim_{x \rightarrow \infty} x^{k} = \infty\) for every positive constant \(k\), we can apply L'Hôpital's rule.

Step 2: Find the derivatives of the numerator and denominator

First, differentiate \(\ln x\) with respect to \(x\), resulting in \(\frac{1}{x}\). Then, differentiate \(x^k\) with respect to \(x\), resulting in \(k\cdot x^{k-1}\).

Step 3: Apply L'Hôpital's Rule to the limit

Using L'Hôpital's Rule, we will now find \(\lim_{x \rightarrow \infty} \frac{\frac{1}{x}}{k\cdot x^{k-1}}\).

Step 4: Simplify the limit

The limit can be simplified as \(\lim_{x \rightarrow \infty} \frac{1}{k\cdot x^{k}}\).

Step 5: Determine the new limit

Since \(\lim_{x \rightarrow \infty} \frac{1}{k\cdot x^{k}} = \frac{1}{\infty} = 0\), we have proved that \(\lim _{x \rightarrow \infty} \frac{\ln x}{x^{k}}=0\) for every positive constant \(k\).

Key Concepts

Limits of Functions

Limits help us understand the behavior of functions as they approach specific points or infinity. It's about what happens to a function's value as the variable gets closer to a particular number or trends to infinity.
In calculus, limits can be used to describe the continuity or discontinuity of functions as well as the potential end behavior of graphs. For example, in the given exercise, we analyzed the limits as \(x\) tends towards infinity.

• The expression \(\lim _{x \rightarrow \infty} \frac{x^k}{e^x} = 0\) indicates that a power function's growth is outpaced by the exponential function as \(x\) increases.
• Similarly, \(\lim _{x \rightarrow \infty} \frac{\ln x}{x^k} = 0\) shows us that the logarithmic growth of a function is too slow to keep up with even the simplest quadratic or higher power functions as \(x\) increases.

Limits are a foundational tool in calculus, which helps in examining how functions behave at boundaries or extremes.

Exponential Growth

Exponential functions express really rapid growth or decay. The most common exponential function \(e^x\), where \(e\) is the base of natural logarithms, showcases exponential growth perfectly.
In the problem, we saw that \(e^x\) increases faster than any power function \(x^k\). This means:

• In a scenario where \(x\) keeps growing, \(e^x\) will rise faster than \(x^2\), \(x^3\), and so on.
• This property is incredibly powerful in mathematics and real-world applications, such as finance for compound interest calculations or biology for population growth models.

Exponential growth leads functions to become incredibly large very quickly.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. The natural logarithm denoted by \(\ln x\) gives the time required for any exponential growth to reach a certain level.
Logarithms grow very slowly, especially compared to power or exponential functions. In the exercise, we proved that \(\ln x\) increases much slower than \(x^k\) for any positive constant \(k\).

• Logarithmic functions are commonly used in calculations dealing with growth rates, such as the Richter scale for earthquake magnitudes or the pH scale in chemistry.

With \(\ln x\), even if \(x\) becomes large, the growth is tempered and kept in check, meaning \(\ln x\) can only increase slowly.

Power Functions

Power functions take the form \(x^k\), where \(k\) is any real number. They can model a variety of growth patterns depending on the exponent \(k\).
When \(k\) is positive, the function \(x^k\) denotes polynomial growth, which is significant but predictable. In contrast to exponential functions, power functions grow considerably slower as demonstrated in the exercise.

• We saw that \(x^k\) could not compete with \(e^x\), showing how power functions are restricted in their rapid growth.
• However, they outpace logarithmic functions like \(\ln x\).

Examples of power functions in the real world include areas such as physics, where they might model phenomena like gravitational forces or speed in mechanics.