Problem 60 - Calculus: Early Transcendentals

Question

Transcendental functions Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=|\ln x|$$

Step-by-Step Solution

Answer

Answer: The end behavior of the function $$f(x) = |\ln x|$$ is as follows: 1. As x approaches positive infinity: $$\lim_{x\to \infty} |\ln x| = \infty$$ 2. As x approaches 0: $$\lim_{x\to 0} |\ln x| = \infty$$ There is a vertical asymptote at $$x=0$$, and the function is undefined for negative x values.

Step 1: Analyze Limit as x Approaches Positive Infinity

As x approaches positive infinity, we want to determine the limit $$\lim_{x\to\infty} |\ln x|$$. Since the natural logarithm of x grows larger as x increases, we can say that the limit is: $$\lim_{x\to\infty} |\ln x| = \infty$$

Step 2: Analyze Limit as x Approaches Negative Infinity

As x approaches negative infinity, the function is undefined because the natural logarithm is only defined for positive x values. Therefore, we don't need to analyze the limit as x approaches negative infinity.

Step 3: Analyze Limit as x Approaches 0

As x approaches 0, we want to determine the limit $$\lim_{x\to 0} |\ln x|$$. Since the natural logarithm of x approaches negative infinity as x approaches 0, the absolute value will make the function approach positive infinity. Thus, the limit is: $$\lim_{x\to 0} |\ln x| = \infty$$

Step 4: Identify Asymptotes

Since the limit of the function approaches infinity as x approaches 0, there is a vertical asymptote at $$x=0$$. There are no horizontal asymptotes since the function goes to infinity as x approaches positive infinity.

Step 5: Provide a Simple Sketch of the Graph

From the analysis above, we know that: 1. The function grows as x increases. 2. There is a vertical asymptote at $$x=0$$. 3. There are no horizontal asymptotes. Based on these observations, we can create a simple sketch of the graph. 1. Draw a vertical dashed line at $$x=0$$ to represent the vertical asymptote. 2. Start the curve near the vertical asymptote and let it grow as x increases. 3. Since the function approaches infinity as x approaches 0 and positive infinity, make sure the curve approaches infinity in those directions. That's the simple sketch of the function $$f(x)=|\ln x|$$.

Key Concepts

End Behavior

End behavior describes how a function behaves as the input values become extremely large or extremely small. For transcendental functions like \(f(x) = |\ln x|\), understanding end behavior helps you visualize how the function behaves at the extremes of its domain.

• As \(x\) approaches positive infinity, the logarithmic function \(\ln x\) grows larger and larger. Therefore, the absolute value \(|\ln x|\) also increases indefinitely. This tells us that \(f(x)\) heads towards positive infinity, meaning it has an end behavior that moves upwards as \(x\) grows larger.


• Conversely, as \(x\) approaches zero from the right, \(\ln x\) approaches negative infinity. When we take the absolute value, \(|\ln x|\) becomes positively infinite. In this way, the function \(f(x)\) also tends towards positive infinity. So on both ends, the graph of \(|\ln x|\) goes off to infinity, but from different points of origin in the domain.

Limits Analysis

Limits analysis is the process of evaluating the behavior of a function as its inputs approach certain critical values. With \(f(x) = |\ln x|\), we need to look at how the function behaves as \(x\) nears specific points like 0 or infinity.

• For \(\lim_{x \to \infty} |\ln x|\), since \(\ln x\) grows infinitely, the absolute value—\(|\ln x|\)—does as well. As such, the limit is \(\infty\).


• For \(\lim_{x \to 0^+} |\ln x|\), as \(x\) closes in on 0, \(\ln x\) approaches negative infinity. But because of the absolute value, \(|\ln x|\) will actually increase towards positive infinity too. Thus, this limit is also \(\infty\).

This analysis of limits provides crucial information about how the function behaves at extreme ranges and helps identify potential asymptotes.

Vertical Asymptotes

Vertical asymptotes occur in a graph where a function approaches infinity as \(x\) approaches a certain value; this behavior generally indicates that the function is undefined at that specific \(x\) value. For the function \(f(x) = |\ln x|\), the vertical asymptote occurs at \(x = 0\).

• Near this point, the function itself grows very large as \(x\) approaches 0 from the right side, heading towards positive infinity. This indicates the presence of a vertical asymptote at \(x=0\), and is typically depicted as a dashed line on the graph, signaling that the curve will keep extending upwards and closely follow this vertical line without ever touching or crossing it.


• This vertical asymptote is essential for sketching the graph and understanding the function’s behavior near \(x=0\), where \(|\ln x|\) dictates a significant increase as \(x\) diminishes towards zero.