Problem 42 - Calculus: Early Transcendentals

Question

Determine the infinite limit. \( \displaystyle \lim_{x \to 0^+}\left( \frac{1}{x} - \ln x \right) \)

Step-by-Step Solution

Answer

The limit is \( \infty \).

Step 1: Understand the Function

The original function is \( f(x) = \frac{1}{x} - \ln x \). As \( x \to 0^+ \), both \( \frac{1}{x} \) and \( \ln x \) approach infinity. We need to determine which term dominates.

Step 2: Analyze \( \frac{1}{x} \)

As \( x \) approaches \( 0^+ \), the function \( \frac{1}{x} \) goes to positive infinity. This term grows without bound as \( x \) gets closer to zero from the positive side.

Step 3: Analyze \( \ln x \)

As \( x \) approaches \( 0^+ \), \( \ln x \) goes to negative infinity. This is because the natural logarithm of a number between 0 and 1 is negative and decreases without bound as the input approaches zero.

Step 4: Compare Growth Rates

To determine the behavior of \( f(x) = \frac{1}{x} - \ln x \), we compare the rates at which \( \frac{1}{x} \) approaches infinity and \( \ln x \) approaches negative infinity. As \( x \to 0^+ \), \( \frac{1}{x} \) grows much faster than \( \ln x \) decreases, which means \( \frac{1}{x} \) dominates the expression.

Step 5: Determine the Limit

Since \( \frac{1}{x} \) dominates and becomes infinite much faster than \( \ln x \), the entire expression \( \frac{1}{x} - \ln x \) approaches positive infinity. Thus, the limit is \( \infty \).

Key Concepts

Dominance of Terms

When evaluating limits that involve the subtraction or addition of different functions, it's key to understand which term "dominates". This means the term that grows faster will impact the final value of the expression the most. In our exercise, we looked at the expression \( f(x) = \frac{1}{x} - \ln x \). It's a situation where each component behaves differently as \( x \to 0^+ \).

• \( \frac{1}{x} \) approaches infinity.
• \( \ln x \) approaches negative infinity.

Here, the dominance of terms is evaluated by comparing how quickly each term approaches its infinite behavior. Since \( \frac{1}{x} \) increases rapidly and \( \ln x \) decreases at a slower pace, we conclude that \( \frac{1}{x} \) is the dominant term, determining the expression's behavior.

Growth Rates

The growth rate of a function describes how quickly its value changes as the input varies. When dealing with limits involving infinity, it's important to know which functions grow more quickly. In the given limit problem, we need to compare the growth rates of \( \frac{1}{x} \) and \( \ln x \) as \( x \to 0^+ \).

• As \( x \to 0^+ \), \( \frac{1}{x} \) skyrockets to infinity, growing faster than \( \ln x \).
• Conversely, \( \ln x \) drifts to negative infinity more slowly than \( \frac{1}{x} \) increases to positive infinity.

This comparison of growth rates helps to establish that \( \frac{1}{x} \) has a larger influence on the overall limit, pushing \( \frac{1}{x} - \ln x \) towards positive infinity.

Natural Logarithm Behavior

The natural logarithm, denoted \( \ln x \), behaves uniquely around the point where \( x \to 0^+ \). Understanding this behavior helps explain why \( \ln x \) approaches negative infinity:

• For values of \( x \) between 0 and 1, \( \ln x \) gives a negative result.
• As \( x \) approaches zero from the positive side, \( \ln x \) keeps decreasing without bound and becomes increasingly negative.

While the natural logarithm tends towards \(-\infty\), its rate of decrease is slower compared to the rapid increase of \( \frac{1}{x} \), establishing its secondary role in the expression \( \frac{1}{x} - \ln x \). Understanding \( \ln x \)'s behavior is key to seeing why it doesn't dominate the limit's outcome.

Positive Infinity

In the context of limits, positive infinity describes the concept of values increasing beyond any finite limit. It is represented by \( \infty \) and indicates a function's growth without bounds. In our case:

• \( \frac{1}{x} \) tends towards positive infinity as \( x \to 0^+ \).
• By comparison, since its growth is much faster than \( \ln x\)'s decrease, the expression \( \frac{1}{x} - \ln x \) also approaches positive infinity.

When a dominant term such as \( \frac{1}{x} \) overpowers opposing terms like \( \ln x \), the sum drives the limit toward positive infinity. This understanding is crucial in analyzing how expressions behave and determine their infinite limits.