Problem 37 - Calculus of a Single Variable

Question

In Exercises 37–40, find the limit. $$ \lim _{x \rightarrow 3^{+}} \ln (x-3) $$

Step-by-Step Solution

Answer

The limit of \(\ln(x-3)\) as \(x\) approaches 3 from the right is \(-\infty\).

Step 1: Analyze the function

Before applying any limit properties, check the behaviour of the function at \(x = 3\). As \(x\) approaches 3 from the right, \(x-3\) becomes very close to 0 but is still positive, hence \(\ln(x-3)\) approaches negative infinity.

Step 2: Apply Limit

Based on the observation made in step 1, we can conclude that the limit as \(x\) approaches 3 from the right, for the function \(\ln(x-3)\), is negative infinity.

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln(x)\), is a fundamental mathematical function. It is the logarithm to the base \(e\), where \(e\) is a mathematical constant approximately equal to 2.71828. Natural logarithms are very common in calculus, especially when dealing with limits and growth problems.

The function \(\ln(x)\) is only defined for \(x > 0\). This is because you cannot take the logarithm of a non-positive number within the real number system. An important characteristic of the logarithm function is its behavior near zero. As \(x\) approaches zero from the positive side, \(\ln(x)\) approaches negative infinity. This is crucial to understanding the exercise given, where \(\ln(x-3)\) behaves similarly as \((x-3)\) approaches zero.

Behavior of Functions

Understanding how functions behave as they approach specific values is essential in calculus. This behavior tells us a lot about limits and continuity.

In the context of the given exercise, we focused on how the function \(\ln(x-3)\) behaves as \(x\) nears 3 from the right. As \(x\) approaches 3, the expression \(x-3\) becomes very small but remains positive.

- Because log functions can only take positive inputs, \(\ln(x-3)\) is defined and tends to negative infinity as the input nears zero.- This is an example of how understanding the behavior and domain of a function can help determine the limit.Recognizing this specific behavior can help make predictions about the outcomes for similar logarithmic functions.

One-Sided Limits

A one-sided limit considers the value that a function approaches as the input approaches a specific point from one side, either from the left or the right. In the given exercise, we are dealing with a right-hand limit, denoted as \(\lim_{x \to 3^+}\).

One-sided limits help clarify the behavior of functions at points where the function might not be defined or continuous. They can also simplify problems where the direction of approach affects the limit.- For \(\ln(x-3)\), the function is not defined at \(x = 3\), but it is for values just greater than 3, leading to a consideration of the limit as \(x\) approaches 3 from the right—the positive side.- These limits are crucial in cases where the function behaves radically differently from different sides of a point.

By understanding one-sided limits, we gain a powerful tool to analyze the function's behavior near points of discontinuity or undefined values.