Problem 15 - Calculus Early Transcendentals

Question

\(15-20\) Explain why the function is discontinuous at the given number a. Sketch the graph of the function. \(f(x)=\ln |x-2| \quad a=2\)

Step-by-Step Solution

Answer

The function is discontinuous at \( x = 2 \) because it is undefined at this point, causing a vertical asymptote.

Step 1: Define the Function

The function given is \( f(x) = \ln |x-2| \). To understand where the function might be discontinuous, we'll analyze its behavior at the point \( a = 2 \).

Step 2: Analyze Domain of the Function

The domain of \( f(x) = \ln |x - 2| \) is all \( x \) such that \( |x - 2| > 0 \). This means \( x eq 2 \). Therefore, the function is not defined exactly at \( x = 2 \).

Step 3: Check Limit from the Left

To check continuity, we examine the left-hand limit of the function as \( x \to 2^- \). As \( x \to 2^- \), \( |x-2| \to 0^+ \) and \( \ln |x-2| \to -\infty \).

Step 4: Check Limit from the Right

Similarly, examine the right-hand limit as \( x \to 2^+ \). Again, \( |x-2| \to 0^+ \) and \( \ln |x-2| \to -\infty \).

Step 5: Compare Limits to Check Continuity

For continuity at \( x = 2 \), the function must be defined at \( x = 2 \), and the left-hand and right-hand limits must be equal and equal to \( f(2) \). Since \( f(2) \) is undefined and both limits approach \(-\infty\), the function is discontinuous at \( x = 2 \).

Step 6: Sketch the Graph

The function is drawn with a vertical asymptote at \( x = 2 \). As \( x \) approaches 2 from either side, \( f(x) \) decreases towards \(-\infty\). Away from \( x = 2 \), the graph resembles typical logarithmic behavior for \(|x-2|>0\).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. The function \( f(x) = \ln(x) \) denotes the natural logarithm of \( x \), which is the power to which the base \( e \) (approximately 2.718) must be raised to yield \( x \). When dealing with expressions like \( \ln |x-2| \), the behavior of the function is determined by the absolute value of \( x-2 \), ensuring the input to the logarithm is positive. - This is because the logarithm is undefined for non-positive numbers; that's why \( \ln 0 \) and \( \ln \text{(negative numbers)} \) don't exist.- The transformation using absolute value changes the function only in terms of where it can be evaluated, not how the logarithm behaves. The graph of a simple logarithmic function, like \( \ln(x) \), has a vertical asymptote at \( x = 0 \) because it approaches negative infinity as \( x \) nears zero from the positive side. Similarly, \( \ln |x-2| \) has an asymptotic behavior at the points where \( |x-2| = 0 \), particularly at \( x = 2 \). This contributes to the discontinuity we observe in some graphs.

Limits and Continuity

Continuity of a function at a point requires three main conditions:

• The function must be defined at that point.
• The left-hand limit and right-hand limit as you approach the point must exist.
• The limits from both sides must equal the function's value at that point.

For \( f(x) = \ln |x-2| \), the function is not defined exactly at \( x = 2 \), because \( \ln(0) \) is undefined. This alone breaches one of the key conditions of continuity. Additionally, when approaching from either side of 2, the function heads towards \(-\infty\), causing both left and right-hand limits to exist but not equal to each other or a finite value. Since the function isn't defined at \( x = 2 \) and both limits are equal to \(-\infty\), it highlights a clear point of discontinuity. A discontinuous function exhibits breaks, jumps, or asymptotes in its graph, as seen here.

Domain of a Function

The domain of a function is the set of all possible input values (\( x \) values) for which the function is defined. For a logarithmic function such as \( f(x) = \ln |x-2| \), the domain encompasses all \( x \) that keep \( |x-2| > 0 \).- This requirement stems from the fact that a logarithmic expression is undefined for zero and negative inputs.- Thus, the absolute value ensures that the part inside \( \ln \) is positive, leading to \( x eq 2 \) as a restriction.Consequently, this function's domain includes all real numbers except \( x = 2 \). When examining the function's graph, this results in a vertical asymptote precisely at the point where the domain restriction occurs, making \( x = 2 \) a critical point of discontinuity. Understanding the domain is crucial for evaluating where the logarithmic function will behave regularly and where it will show signs of discontinuity.