Problem 23 - APEX Calculus

Question

Evaluate the given limit. $$ \lim _{x \rightarrow 0} \ln x $$

Step-by-Step Solution

Answer

The limit is \(-\infty\).

Step 1: Understand the function and the limit

We need to evaluate the limit of the natural logarithm function, \(\ln(x)\), as \(x\) approaches 0. The function is only defined for \(x>0\). Hence, we will consider the behavior of the function as \(x\) approaches 0 from the right (i.e., \(x \to 0^+\)).

Step 2: Analyze the behavior of \(\ln(x)\) as \(x \to 0^+\)

The natural logarithm function \(\ln(x)\) is defined and continuous for positive \(x\). As \(x\) gets smaller and moves towards zero from the right, the function \(\ln(x)\) decreases without bound. This implies that \(\ln(x)\) approaches negative infinity.

Step 3: Evaluate the limit

Considering the behavior described in Step 2, the limit \(\lim_{x \to 0^+} \ln(x) = -\infty\). This is because as \(x\) gets smaller (but positive), \(\ln(x)\) continues to decrease indefinitely.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is a special logarithm that has a base of \( e \), where \( e \approx 2.71828 \). It is one of the most important functions in mathematics due to its applications in growth and change models.

Properties of the natural logarithm include:

• \( \ln(1) = 0 \) because \( e^0 = 1 \).
• For \( x > 1 \), \( \ln(x) > 0 \).
• For \( 0 < x < 1 \), \( \ln(x) < 0 \).
• It is defined only for positive \( x \), which means the domain is \( (0, \infty) \).

Understanding \( \ln(x) \) at values close to zero helps us evaluate limits and understand the behavior of functions at boundary points like \( x \to 0^+ \).

Negative Infinity

Negative infinity, represented as \( -\infty \), is used in mathematics to describe values that decrease without bound. It is a concept rather than a specific number, helping to understand situations where functions continuously decrease as an input approaches a certain point.

In the context of limits, saying a function approaches \( -\infty \) means that as the input becomes closer to a specific value, the function’s output decreases indefinitely. For the function \( \ln(x) \), as \( x \to 0^+ \), the log continues to drop, moving toward \( -\infty \). This behavior is crucial for analyzing real-world phenomena, such as decay processes, which might resemble a logarithmic drop-off.

Continuous Function

The term "continuous function" refers to a function that does not have any abrupt changes in value, meaning there are no breaks, jumps, or holes anywhere in the domain.

For a function \( f(x) \) to be continuous at a point \( c \), the following must hold:

• \( f(c) \) is defined.
• The limit \( \lim_{x \to c} f(x) \) exists.
• \( \lim_{x \to c} f(x) = f(c) \).

The function \( \ln(x) \) is continuous for all \( x > 0 \), which means it doesn’t have any sudden jumps or undefined points in this domain. When we evaluate the limit of \( \ln(x) \) as \( x \to 0^+ \), we take advantage of its continuity to better understand its behavior as it moves toward this boundary of defined values.

Behavior Asymptotes

Asymptotic behavior describes how a function behaves as the input approaches a specific boundary or point, often leading to very large or small values. It helps to predict the long-term behavior of complex functions.

In mathematical terms, an asymptote is a line that a graph approaches but never touches. For \( \ln(x) \), as \( x \to 0^+ \), the function heads towards \( -\infty \). Here, the vertical line \( x = 0 \) acts as a vertical asymptote.

Understanding asymptotic behavior is key in calculus and higher mathematics because it helps us predict how functions behave at boundaries and end points. This understanding ties into evaluating the limit \( \lim_{x \to 0^+} \ln(x) \) and recognizing the asymptote along \( x = 0 \).