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Chapter 2: Problem 43

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

Short Answer

Expert verified
As \(x\) approaches 0 from the positive side, \(\ln x\) goes to \(-\infty\).

Step by step solution

01

Understanding the Definition of the Limit

To show that the limit is \(-\infty\), we are essentially proving that as \(x \rightarrow 0^{+}\), the natural logarithm \(\ln x\) decreases without bound. This involves showing that for every large negative number \(L\), there exists a \(\delta > 0\) such that whenever \(0 < x < \delta\), \(\ln x < L\).
02

Choose a Large Negative Number

Let \(L\) be any large negative number. Our goal is to find a \(\delta > 0\) such that whenever \(0 < x < \delta\), the inequality \(\ln x < L\) holds. This essentially means we need to determine the condition on \(x\) for which the natural log becomes smaller than our chosen \(L\).
03

Solving the Inequality ln(x) < L

Start by solving the inequality \(\ln x < L\). Exponentiate both sides to eliminate the logarithm: \((e^{\ln x}=e^L)\), giving us \(x < e^L\). For \(L < 0\), \(e^L\) is a positive number less than 1. This implies that if we choose \(\delta = e^L\), for any \(0 < x < \delta\), \(\ln x < L\) holds true.
04

Conclusion of the Limit

Since for any large negative number \(L\), we can always choose \(\delta = e^L\) such that the inequality \(\ln x < L\) is satisfied whenever \(0 < x < \delta\), we have shown that as \(x\) approaches 0 from the positive side, \(\ln x\) decreases without bound. Thus, \(\lim _{x \rightarrow 0^{+}} \ln x=-\infty\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Natural Logarithm
The natural logarithm, often denoted as \( \ln x \), is a special logarithmic function with base \( e \), where \( e \approx 2.71828 \). It's called "natural" because it appears frequently in mathematics and natural sciences.
The natural logarithm has unique properties:
  • It is undefined for negative numbers and zero.
  • \( \ln 1 = 0 \) because \( e^0 = 1 \).
  • As \( x \) increases, \( \ln x \) also increases.
  • The domain of \( \ln x \) is \( x > 0 \).
The inverse operation is exponential, meaning if \( y = \ln x \), then \( x = e^y \).
In the context of limits, understanding how \( \ln x \) behaves as \( x \) approaches zero from the right (\( x \rightarrow 0^{+} \)) is crucial. Here, the logarithmic function decreases rapidly towards negative infinity, as shown in the provided step-by-step solution.
Limit at a Point
The concept of a "limit at a point" is fundamental in calculus. It describes the behavior of a function as the input approaches a particular value. This involves examining the function values as they get arbitrarily close to a target point.
  • If as \( x \to c \), \( f(x) \) approaches a specific number \( L \), we say the limit of \( f(x) \) as \( x \) approaches \( c \) is \( L \).
  • Notation: \( \lim_{x \to c} f(x) = L \).
  • Limits can exist even if \( f(x) \) is not defined at \( c \).
For the limit \( \lim_{x \to 0^{+}} \ln x = -\infty \), the idea is that \( \ln x \) becomes negatively infinite as \( x \) nears zero, signifying no bound in negative direction. This is supported by step-by-step inequality solutions from the original exercise.
Exponential Function
The exponential function is a mathematical function represented by \( e^x \), where \( e \) is Euler's number, approximately 2.71828.
It provides the inverse function to the natural logarithm. In other words, if \( y = e^x \), then the natural logarithm gives \( \ln y = x \).
  • Rapid growth: As \( x \) increases, \( e^x \) grows exponentially.
  • Always positive: Regardless of \( x \), \( e^x > 0 \).
  • The derivative of \( e^x \) is \( e^x \) itself, showing its self-replicating rate of change.
The exponential function is essential in the original example because it helps solve inequalities and understand exponential decay or growth. As shown, when solving \( \ln x < L \), exponentials clarify the threshold at which \( x \) can satisfy the condition, thus providing insights into the behavior of logarithmic functions near extreme bounds.

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