Problem 13 - Calculus - AP Edition

Question

Find the limits. $$\lim _{x \rightarrow+\infty} \frac{\ln x}{x}$$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understanding the Limit

We need to find the limit of the function \( \lim_{x \to +\infty} \frac{\ln x}{x} \). As \( x \) approaches infinity, both \( \ln x \) and \( x \) get larger, but \( x \) grows at a different rate compared to \( \ln x \). This step sets up the expectation that \( x \) may grow faster than \( \ln x \).

Step 2: Applying L'Hospital's Rule

Since direct substitution leads to an indeterminate form \( \frac{\infty}{\infty} \), we use L'Hospital's Rule, which states that if a limit leads to an indeterminate form, \( \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \), provided the new limit exists. We apply this rule here.

Step 3: Differentiating Numerator and Denominator

Differentiate the numerator \( \ln x \) to get \( \frac{1}{x} \). Differentiate the denominator \( x \) to get \( 1 \). Now, the limit becomes: \( \lim_{x \to +\infty} \frac{\frac{1}{x}}{1} = \lim_{x \to +\infty} \frac{1}{x} \).

Step 4: Evaluating the New Limit

Now we evaluate the new limit: \( \lim_{x \to +\infty} \frac{1}{x} \). As \( x \) approaches infinity, \( \frac{1}{x} \) approaches \( 0 \), since the numerator remains 1 while the denominator grows indefinitely.

Step 5: Concluding the Result

The limit of the original function is \( 0 \). Thus the final result for the problem is \( \lim_{x \to +\infty} \frac{\ln x}{x} = 0 \). This indicates that the function's rate of growth dwindles towards zero as \( x \) grows without bound.

Key Concepts

Understanding Limits

Limits help us determine the behavior of a function as its input gets closer to a certain value. In the case of this exercise, we are interested in finding the limit of the function \( \lim_{x \to +\infty} \frac{\ln x}{x} \), where \( x \) approaches infinity. This tells us how the fraction \( \frac{\ln x}{x} \) behaves as \( x \) gets larger and larger.

As \( x \to +\infty \), both \( \ln x \) and \( x \) get larger. However, they do not grow at the same rate. This discrepancy is where understanding limits becomes crucial. Although both functions seem to get large, determining which grows faster is key to solving the limit problem. Here, intuition might suggest that \( x \) grows faster than \( \ln x \), leading to a smaller resulting ratio.

Exploring Indeterminate Forms

Indeterminate forms arise when direct substitution into a limit results in expressions like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). In this exercise, using direct substitution for \( \lim_{x \to +\infty} \frac{\ln x}{x} \) leads to \( \frac{\infty}{\infty} \). Such forms don't provide enough information to determine the limit right away.

L'Hospital's Rule helps us tackle these situations. It allows us to differentiate both the numerator and the denominator separately until we can resolve the indeterminate form and find the limit. This method works here, transforming the problem into a simpler one, clearly illustrating the power and utility of L'Hospital's Rule in calculus.

Comparing Rates of Growth

The rate of growth is essential to understanding the behavior of functions as their input changes, particularly in the context of limits. With \( \ln x \) and \( x \), comparing their growth rates helps us predict the outcome of the limit. In our problem, while both \( \ln x \) and \( x \) grow as \( x \to +\infty \), \( x \) grows significantly faster.

After applying L'Hospital's Rule and simplifying the expression to \( \lim_{x \to +\infty} \frac{1}{x} \), it becomes evident that \( \frac{1}{x} \) approaches 0 as \( x \to +\infty \). This calculation shows how fast the function diminishes in contrast to the much slower growth of the logarithmic function. Such assessments of growth rates are pivotal for accurately determining limits and understanding the overarching behavior of functions in calculus.