Problem 43 - Calculus Early Transcendentals

Question

Prove that $$\lim _{x \rightarrow 0^{+}} \ln x=-\infty$$

Step-by-Step Solution

Answer

As \( x \rightarrow 0^{+} \), \( \ln x \) decreases indefinitely, thus \( \lim _{x \rightarrow 0^{+}} \ln x = -\infty \).

Step 1: Understand the Limit Expression

The expression \( \lim _{x \rightarrow 0^{+}} \ln x = -\infty \) means we need to show that as \( x \) approaches 0 from the positive side, the value of \( \ln x \) decreases without bound (becomes infinitely negative). This involves evaluating how \( \ln x \) behaves as \( x \) gets closer to 0.

Step 2: Analyze the Behavior of the Natural Logarithm Function

The natural logarithm function \( \ln x \) is only defined for positive values of \( x \). As \( x \) approaches positive values closer to 0, \( \ln x \) drops sharply. Observationally, we can note that \( \ln x \) becomes more and more negative as \( x \) decreases towards 0.

Step 3: Argue using the Properties of Natural Logarithm

For \( 0 < x < 1 \), the value of \( \ln x \) is negative because the natural log of numbers between 0 and 1 is negative. More formally, notice that \( \ln(x) \) is the inverse of the exponential function; therefore as \( x \rightarrow 0^{+} \), \( e^{\ln x} \rightarrow 0 \). Both \( \ln(1) = 0 \) and as \( x \) moves closer to 0, the slope of \( \ln x \) causes it to drop infinitely negatively.

Step 4: Apply a Formal Limit Approach

Consider any large negative number \( N \). We need to show that there exists a positive \( \delta \) such that for all \( 0 < x < \delta \), \( \ln x < N \). Take \( \delta = e^N \). For any \( 0 < x < \delta \), we have \( 0 < x < e^N \) implying \( \ln x < N \), since \( \ln(e^N) = N \). This confirms that \( \ln x \rightarrow -\infty \) as \( x \rightarrow 0^{+} \).

Key Concepts

Natural Logarithm

The natural logarithm, represented as \( \ln x \), is a fundamental mathematical function widely used in calculus and many fields of science and engineering. Let's break this down further to understand its special characteristics:

• Definition and Domain: The natural logarithm is only defined for positive values of \( x \). This means \( \ln x \) can be used only when \( x > 0 \).
• Inverse Property: The natural logarithm is the inverse of the exponential function (i.e., \( e^{y} = x \) implies \( y = \ln x \)). This relationship is key to understanding the behavior of logarithmic functions.
• Slope and Growth: For values of \( x \) between 0 and 1, \( \ln x \) is negative, and as \( x \) decreases towards zero, \( \ln x \) decreases rapidly. This results in a steep negative slope as you move leftward towards zero.

Each of these aspects plays a critical role in exploring limits and understanding the behavior of functions as \( x \) changes.

Limits at Infinity

Calculus often deals with limits, which describe the behavior of a function as \( x \) approaches a certain value or infinity. Let's dive into what limits at infinity entail:

• Concept: The notation \( \lim_{x \to \infty} \) or \( \lim_{x \to 0^{+}} \) indicates we are interested in the behavior of a function as \( x \) becomes very large, or very close to zero, respectively.
• Natural Logarithm and Infinity: For the natural logarithm \( \ln x \), as \( x \to \infty \), \( \ln x \to \infty \). Conversely, as \( x \to 0^{+} \), \( \ln x \) goes to \(-\infty\).

Consider the limit \( \lim_{x \to 0^{+}} \ln x = -\infty \). This implies that as \( x \) gets tinier and tinier, closer to zero from the positive side, the values of \( \ln x \) become hugely negative, without bound.

Behavior as x Approaches Zero

Understanding how functions behave as \( x \) approaches zero is essential in calculus. It gives insight into the limiting behavior and helps solve many mathematical problems.

• Approaching Zero from Positive Side: When we examine the limit \( \lim_{x \to 0^{+}} \ln x \), we focus on \( x \) reducing towards 0 from the positive side.
• Function Behavior: As \( x \) approaches zero, \( \ln x \) drops sharply towards negative infinity since the natural log of numbers less than one is negative.
• Mathematical Proof: By picking any large negative number \( N \) and setting \( \delta = e^N \), we prove that for any \( x < \delta \), \( \ln x \) becomes smaller than \( N \). Hence, it drops beyond any bound as \( x \to 0^{+} \).

These steps illustrate how \( \ln x \) behaves dramatically as it approaches zero, providing a clear mathematical foundation for its limit behavior.