Question

\(\mathrm{T} / \mathrm{F}: \lim _{x \rightarrow 1} \ln x=0 .\) Use a theorem to defend your answer.

Short answer

True, \(\lim_{x \rightarrow 1} \ln x = 0\) by the continuity of the logarithm at \(x=1\).

Step-by-step solution

Understanding the Problem

We need to determine if the statement \(\lim_{x \rightarrow 1} \ln x = 0\) is true or false. This involves evaluating the limit of the natural logarithm of \(x\) as \(x\) approaches 1.

Recall the Limit Definition

The limit \(\lim_{x \rightarrow c} f(x) = L\) means that as \(x\) gets arbitrarily close to \(c\), \(f(x)\) gets arbitrarily close to \(L\). We need to use this to check if \(\ln x\) approaches 0 as \(x\) approaches 1.

Apply the Logarithm Property at Limit

Recall that \(\ln(1) = 0\). As \(x\) approaches 1, \(\ln(x)\) approaches \(\ln(1)\), which equals 0. Therefore, the function \(\ln(x)\) is continuous at \(x = 1\).

Use Continuity Theorem

By the properties of continuous functions, \(f(x) = \ln(x)\) is continuous at \(x = 1\). Therefore, \(\lim_{x \rightarrow 1} \ln x = \ln(1) = 0\). For continuous functions, \(\lim_{x \rightarrow c} f(x) = f(c)\).

Conclusion

Since \(\ln(x)\) is continuous at \(x = 1\), and \(\ln(1) = 0\), the limit \(\lim_{x \rightarrow 1} \ln x = 0\) is correct. The statement is true based on the continuity theorem.

Key concepts

Continuity Theorem

The Continuity Theorem is a fundamental concept that helps explain how limits work for continuous functions. A function is continuous at a point, say \(x = c\), if the following holds:

• The function \(f(x)\) is defined at \(x = c\).
• \( \lim_{x \to c} f(x) \) exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) is equal to \(f(c)\).

In simpler terms, a continuous function has no breaks, jumps, or holes at the point of interest. This means that as \(x\) gets closer and closer to \(c\), the function's value \(f(x)\) gets closer and closer to \(f(c)\).

In the given exercise, the function \(f(x) = \ln(x)\) is continuous at \(x = 1\). This is because the natural logarithm function \(\ln(x)\) is defined everywhere for \(x > 0\). Therefore, we can apply the Continuity Theorem to conclude that \(\lim_{x \to 1} \ln(x) = \ln(1) = 0\). By using the properties of continuity, evaluating such limits becomes straightforward.

Natural Logarithm

The Natural Logarithm, often denoted as \( \ln(x) \), is a logarithm with the base \(e\), where \(e\) is approximately 2.71828. It's one of the most common types of logarithms used in mathematics.

Think of \( \ln(x) \) as answering the question: "To what power must we raise \(e\) to get \(x\)?" Mathematically, the natural logarithm of a number \(x\) can be expressed as:

• \( e^{\ln(x)} = x \)
• \( \ln(e) = 1 \)
• \( \ln(1) = 0 \)

Understanding these properties helps evaluate limits involving \( \ln(x) \). In the exercise, \( \ln(x) \) approaching \(1\) leads us to \( \ln(1) \), which simplifies to \(0\). This property of \( \ln(1) = 0 \) plays a crucial role in confirming the limit of \( \ln(x) \) as \(x\) approaches \(1\).

Recognizing the role \( e \) and the natural logarithm play in calculus is essential, especially when evaluating limits and considering the continuity of functions.

Limit Evaluation

Limit Evaluation is a key technique in calculus that helps us understand the behavior of a function as it approaches a certain point. For any function \(f(x)\), we want to know what value it gets closer to as \(x\) approaches a particular value \(c\). This is expressed as \(\lim_{x \to c} f(x)\).

Evaluating limits involves checking:

• What value does \(f(x)\) approach as \(x\) gets dangerously close to \(c\)?
• Does \(f(x)\) reach a specific number, or does it tend to infinity, or perhaps not settle on a number at all?

In the given problem, we're checking the limit \(\lim_{x \to 1} \ln x = 0\). Here, using the function's continuity at \(x = 1\), we directly evaluate this limit.

By understanding the definition of limits and the behavior of the natural logarithm, we confirm that as \(x\) approaches \(1\), \( \ln(x) \) indeed approaches \(0\). Thanks to the concept of limits, we can rigorously conclude and understand changes in function values around certain points.