Problem 6 - Thomas Calculus

Question

Which of the following functions grow faster than \(\ln x\) as \(x \rightarrow \infty ?\) Which grow at the same rate as \(\ln x ?\) Which grow slower? $$ \begin{array}{ll}{\text { a. } \log _{2}\left(x^{2}\right)} & {\text { b. } \log _{10} 10 x} \\ {\text { c. } 1 / \sqrt{x}} & {\text { d. } 1 / x^{2}}\\\\{\text { e. } x-2 \ln x} & {\text { f. } e^{-x}} \\ {\text { g. } \ln (\ln x)} & {\text { h. } \ln (2 x+5)}\end{array} $$

Step-by-Step Solution

Answer

a, b, h grow at the same rate as \(\ln x\); e grows faster; c, d, f, g grow slower.

Step 1: Verify Growth of log_2(x^2) vs ln(x)

Consider the function \(f(x) = \log_2(x^2)\). We can express it in terms of the natural logarithm: \(\log_2(x^2) = 2 \log_2(x) = \frac{2 \ln x}{\ln 2}\). Since it is a constant multiple of \(\ln x\), \(\log_2(x^2)\) grows at the same rate as \(\ln x\) as \(x \to \infty\).

Step 2: Verify Growth of log_10(10x) vs ln(x)

Consider the function \(f(x) = \log_{10}(10x)\). Using properties of logarithms: \(\log_{10}(10x) = \log_{10}(10) + \log_{10}(x) = 1 + \log_{10}(x) = 1 + \frac{\ln x}{\ln 10}\). This means \(\log_{10}(10x)\) grows at the same rate as \(\ln x\) as \(x \to \infty\).

Step 3: Verify Growth of 1/sqrt(x) vs ln(x)

For \(f(x) = 1/\sqrt{x}\), as \(x \to \infty\), \(1/\sqrt{x}\) approaches 0. Thus, \(1/\sqrt{x}\) grows slower than \(\ln x\).

Step 4: Verify Growth of 1/x^2 vs ln(x)

Consider \(f(x) = 1/x^2\). As \(x \to \infty\), \(1/x^2\) also approaches 0, so it grows slower than \(\ln x\).

Step 5: Verify Growth of x - 2 ln x vs ln(x)

For \(f(x) = x - 2\ln x\), the \(x\) term dominates since its growth is linear, whereas \(\ln x\) is logarithmic. Therefore, \(f(x)\) grows faster than \(\ln x\) as \(x \to \infty\).

Step 6: Verify Growth of e^{-x} vs ln(x)

For \(f(x) = e^{-x}\), as \(x \to \infty\), \(e^{-x}\) approaches 0. Thus, \(e^{-x}\) grows slower than \(\ln x\).

Step 7: Verify Growth of ln(ln x) vs ln(x)

Consider \(f(x) = \ln(\ln x)\). Since \(\ln \) inside the \(\ln\) function grows slower than \(\ln x\), \(\ln(\ln x)\) also grows slower than \(\ln x\) as \(x \to \infty\).

Step 8: Verify Growth of ln(2x+5) vs ln(x)

Consider \(f(x) = \ln(2x+5)\). This function can be rewritten as \(\ln(2(x + 2.5)) = \ln 2 + \ln(x + 2.5)\). As \(x \to \infty\), \(\ln(x + 2.5) \sim \ln x\), so \(\ln(2x+5)\) grows at the same rate as \(\ln x\).

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln x \), is a fundamental concept in mathematics, particularly in calculus and compound interest problems. It is the logarithm to the base \( e \), where \( e \) (approximately 2.718) is an irrational constant known as Euler's number. The natural logarithm has the unique property that it’s the inverse function of the exponential function \( e^x \).

Key properties of the natural logarithm include:

• The logarithm of 1 is zero: \( \ln(1) = 0 \).
• Product rule: \( \ln(ab) = \ln a + \ln b \).
• Quotient rule: \( \ln\left(\frac{a}{b}\right) = \ln a - \ln b \).
• Power rule: \( \ln(a^b) = b \ln a \).
• Derivative: The derivative of \( \ln x \) is \( \frac{1}{x} \).

In many analytic contexts, like comparing growth rates, \( \ln x \) offers insights into the relative intensity of growth across various functions as \( x \to \infty \). Understanding the properties and behavior of \( \ln x \) is crucial for mastering mathematical analysis and solving complex calculus problems.

Rate of Growth

The rate of growth in mathematical functions refers to how quickly a function increases or decreases as its input (usually \( x \)) becomes very large or very small. In the context of functions like \( \ln x \), comparing their growth helps us understand which functions dominate others in the context of size or limits.

For example, when evaluating whether a function grows faster, slower, or at the same rate as \( \ln x \), certain comparisons are typically made:

• Linear growth: Functions like \( f(x) = x - 2 \ln x \) grow faster than \( \ln x \) due to the dominance of the linear term \( x \).
• Logarithmic growth: Functions such as \( \log_2(x^2) \) grow at the same rate as \( \ln x\), since they are scalar multiples of \( \ln x \).
• Sub-linear growth: Functions like \( f(x) = e^{-x} \) or \( f(x) = 1/x^2 \) grow slower than \( \ln x \) since they diminish towards zero as \( x \to \infty \).

Recognizing these patterns is essential for domain applications in calculus or when analyzing algorithms. By understanding how different functions interact and compare in growth, one can make more informed predictions and decisions about their behavior at infinitely large values of \( x \).

Logarithmic Functions

Logarithmic functions, including the natural logarithm, are critical in expressing relationships where changes in the rate of growth tend to slow down as values become extremely large. They are inversely related to exponential functions, providing a bridge to understanding complex systems and their growth patterns.

The general form of a logarithmic function is \( f(x) = \log_b(x) \), where \( b \) is the base of the logarithm. Base \( e \) is a special case, forming the natural logarithm function \( \ln x \), known for its ease of differentiation.

Useful transformations in logarithmic functions often simplify complex expressions into more manageable forms:

• Change of Base Formula: \( \log_b(x) = \frac{\ln x}{\ln b} \), allowing comparison across different bases.
• Logarithm of a Product: Simplifies into a sum: \( \log_b(xy) = \log_b(x) + \log_b(y) \).
• Logarithm of a Quotient: Expressed as a difference: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \).
• Logarithm of a Power: Converts to \( c \cdot \log_b(x) \) if \( x^c \): \( \log_b(x^c) = c \cdot \log_b(x) \).

These properties allow logarithmic functions to precisely model scenarios in science, finance, and information theory, where growth rates are inherently constrained or decelerating.

Limits at Infinity

Limits at infinity involve evaluating the behavior of functions as the variable, \( x \), grows without bound towards positive infinity (\( x \to +\infty \)) or negative infinity (\( x \to -\infty \)). These limits help determine the asymptotic behavior of functions and are valuable to understand how fast or slow a function approaches a certain behavior.

When comparing functions using limits at infinity, certain general observations can be made:

• Logarithmic Functions: Such as \( \ln x \) increase without bound, but at a slower rate compared to polynomial or exponential functions.
• Polynomial Functions: \( x^n \) exhibits much faster growth than \( \ln x \) as \( x \to +\infty \).
• Exponential Functions: \( e^x \) grows significantly faster than both polynomial and logarithmic functions.
• Rational Functions and Reciprocals: \( 1/x^n \rightarrow 0 \) as \( x \to +\infty \), indicating a slower growth compared to \( \ln x \).

Analyzing limits at infinity is essential in calculus to fully grasp how functions behave in extreme conditions, serving as a foundational tool for asymptotic analysis in advanced mathematics and various applied fields.