Problem 48 - Calculus Early Transcendentals

Question

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 that it approaches infinity as x approaches 0 from the positive side and also approaches infinity as x approaches positive infinity. The function has a vertical asymptote at x=0 and has no horizontal asymptotes.

Step 1: Analyze the limit as x approaches 0 from the positive side

We first need to find the limit as x approaches 0 from the positive side: $$\lim_{x\to0^+} |\ln x|$$ Since the natural logarithm of any number less than 1 but greater than 0 is negative, the absolute value will make these values positive. As x approaches 0 from the positive side, the natural logarithm becomes increasingly negative, so the absolute value becomes increasingly positive. Thus, the limit is: $$\lim_{x\to0^+} |\ln x| = \infty$$

Step 2: Analyze the limit as x approaches positive infinity

Next, we need to find the limit as x approaches positive infinity: $$\lim_{x\to\infty} |\ln x|$$ The natural logarithm of any number greater than 1 is positive, hence the absolute value function is just the natural logarithm function itself in this case. As x approaches positive infinity, the natural logarithm also approaches positive infinity. Thus, the limit is: $$\lim_{x\to\infty} |\ln x| = \infty$$

Step 3: Identifying asymptotes

Now, let's identify any asymptotes that may exist. Vertical asymptotes occur at points where the function is undefined or the limit as x approaches a specific value is infinite. In our case, the function is undefined at x=0 and the limit as x approaches 0 from the positive side is infinite, so the function has a vertical asymptote at x=0: Vertical asymptote: x=0 The function's end behavior suggests that it does not have any horizontal asymptotes, as it approaches infinity in both directions.

Step 4: Sketch the graph

Now, we can provide a simple sketch of the graph of the function $$f(x)=|\ln x|$$: 1. The function is undefined at x=0, so draw a vertical asymptote at x=0. 2. As x approaches 0 from the positive side, the function approaches infinity. 3. As x approaches positive infinity, the function approaches infinity. 4. Plot a few points to help visualize the shape of the graph, such as (1,0) (e,1), and (e^2,2). 5. Keep in mind that the graph should lie entirely above the x-axis due to the absolute value function. After drawing the graph, we can see that it starts at the vertical asymptote x=0 and increases as x approaches positive infinity, with no horizontal asymptotes.

Key Concepts

End Behavior

Understanding the end behavior of a function is crucial for sketching its graph and predicting how it behaves as the input values become very large or very small. For the function \( f(x) = |\ln x| \), we're interested in two main scenarios: as \( x \) approaches zero from the positive side, and as \( x \) approaches positive infinity.

- **As \( x \to 0^+ \)**: The logarithm \( \ln x \) becomes largely negative, because the natural log of numbers between 0 and 1 is negative. However, due to the absolute value, \( |\ln x| \) will translate these large negative values into large positive ones. Therefore, the function shoots off towards positive infinity.- **As \( x \to \infty \)**: Here, the natural log \( \ln x \) is simply increasing positively and steadily. Therefore, \( |\ln x| \) behaves just like \( \ln x \) and will also trend upwards towards infinity.In both cases, \( f(x) \) exhibits behavior moving towards infinity, indicating that the graph rises indefinitely on both ends of the spectrum.

Limits

Limits are a fundamental concept when analyzing the behavior of transcendental functions, especially in identifying how a function behaves near certain points. In this exercise, we closely examine limits to explain \( f(x) = |\ln x| \):
- **Limit as \( x \to 0^+ \)**: - The limit \( \lim_{x\to0^+} |\ln x| = \infty \) signifies that as \( x \) gets very close to zero from the positive side, the function's output increases endlessly.- **Limit as \( x \to \infty \)**: - The limit \( \lim_{x\to\infty} |\ln x| = \infty \) indicates that the function keeps growing as \( x \) heads towards infinity.These results provide critical insights into the function's behavior, confirming the drastic rise in output whether \( x \) is very small or very large, without approaching any fixed boundary.

Asymptotes

Asymptotes play a significant role in sketching and understanding the graph of functions where limits tend to infinity. For \( f(x) = |\ln x| \), the notion of asymptotes is primarily concerned with vertical outlooks due to the behavior of the natural logarithm.
- **Vertical Asymptote**: - The function is undefined at \( x = 0 \), and the limit as \( x \) approaches zero from the positive side is infinite. Hence, \( x = 0 \) acts as a vertical asymptote.- **Horizontal Asymptote**: - There is no horizontal asymptote here because the function doesn't approach a finite value as \( x \) moves in either direction. Instead, it always trends positively.Recognizing these asymptotic behaviors aids in accurately drawing the function's graph, depicting how it stretches towards infinity near \( x = 0 \) and in the positive direction, emphasizing the open ends of its domain.