Problem 66 - Calculus for Biology and Medicine

Question

Show that $$ \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}}=0 $$ for any number \(p>0\). This shows that the logarithmic function grows more slowly than any positive power of \(x\) as \(x \rightarrow \infty\).

Step-by-Step Solution

Answer

The limit \( \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}} = 0 \) for any \( p > 0 \).

Step 1: Understanding the Expression

The expression we are working with is \( \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}} \). The goal is to show that this limit equals 0 for any positive value of \( p \). The numerator, \( \ln x \), grows logarithmically, while the denominator, \( x^p \), grows polynomially.

Step 2: L'Hôpital's Rule Setup

Since this is an indeterminate \( \frac{\infty}{\infty} \) form, we can apply L'Hôpital's Rule. L'Hôpital's Rule states that \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \) provided both \( \lim_{x \to c} f(x) = \lim_{x \to c} g(x) = \infty \) or 0. Here, \( f(x) = \ln x \) and \( g(x) = x^p \).

Step 3: Differentiate the Numerator and Denominator

Differentiate \( \ln x \) to get \( f'(x) = \frac{1}{x} \). Differentiate \( x^p \) to get \( g'(x) = p x^{p-1} \).

Step 4: Applying L'Hôpital's Rule

According to L'Hôpital's Rule, the limit \( \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}} \) becomes \( \lim _{x \rightarrow \infty} \frac{\frac{1}{x}}{ px^{p-1}} \). This simplifies to \( \lim _{x \rightarrow \infty} \frac{1}{px^p} \).

Step 5: Simplifying the Limit

Further simplify to \( \lim _{x \rightarrow \infty} \frac{1}{p} \cdot \frac{1}{x^p} \). As \( x \rightarrow \infty \), \( \frac{1}{x^p} \) approaches 0. Thus, the limit equals 0 regardless of the positive value of \( p \).

Key Concepts

L'Hôpital's Rule

When dealing with limits in calculus, you may encounter expressions that result in indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). These forms don't provide useful information about the limit directly. That's where **L'Hôpital's Rule** comes into play. This rule allows us to evaluate such limits by differentiating the numerator and the denominator separately.
Let's see how this works:

• First, confirm that the limit you're trying to solve results in an indeterminate form. In our problem, the expression \( \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}} \) indeed gives \( \frac{\infty}{\infty} \).
• According to L'Hôpital's Rule, we differentiate \( \ln x \) to get \( \frac{1}{x} \), and \( x^p \) to get \( px^{p-1} \).
• The new limit becomes \( \lim _{x \rightarrow \infty} \frac{\frac{1}{x}}{ px^{p-1}} = \lim _{x \rightarrow \infty} \frac{1}{px^p} \).

Applying L'Hôpital's Rule simplifies the limit evaluation process. It transforms complicated expressions into much simpler ones that are easier to solve.

Logarithmic Function Growth

The logarithmic function, often written as \( \ln x \) for natural logarithms, grows in a specific way that is quite different from polynomial functions. **Logarithmic growth** is important to understand, especially when comparing it to other types of growth, like polynomial growth.
Key characteristics of logarithmic growth include:

• Logarithmic functions grow very slowly compared to polynomial functions. While values will continue to increase as \( x \) becomes very large, they do so at a decreasing rate.
• On a graph, the logarithmic function slope gradually becomes less steep as \( x \) increases.
• This characteristic means that as \( x \rightarrow \infty \), \( \ln x \) becomes negligible when compared to polynomials. This is exactly why \( \lim _{x \rightarrow \infty} \frac{\ln x}{x^{p}} = 0 \) regardless of the positive value of \( p \).

Understanding the subtleties of how logarithmic growth compares to other growth types will enrich your overall understanding of calculus concepts.

Polynomial Growth

Polynomial functions, written in the form of \( x^n \), where \( n \) is a positive integer, exhibit **polynomial growth**. These functions show a distinct pattern of increases, especially when compared to slower-growing functions like logarithms.
Some features of polynomial growth include:

• As \( x \rightarrow \infty \), polynomial functions grow indefinitely and at a pace that accelerates as \( n \) (the power of \( x \)) becomes larger.
• The rate at which polynomial functions grow depends on the exponent. A higher power means faster acceleration of the growth rate.
• Compared to logarithmic functions, polynomials outpace them rapidly as \( x \) increases. This is why the expression \( \frac{\ln x}{x^{p}} \rightarrow 0 \) as \( x \rightarrow \infty \). The denominator, being a polynomial, tends to grow much faster compared to the logarithmic numerator.

Recognizing the differences in growth types, particularly between polynomials and logarithms, is crucial in calculus. It helps explain why certain limits evaluate to zero, as in this exercise.