Problem 43 - Calculus: Early Transcendentals

Question

(a) For \( f(x) = \frac{x}{\ln x} \) find each of the following limits. (i) \( \displaystyle \lim_{x \to 0^+} f(x) \) (ii) \( \displaystyle \lim_{x \to 1^-} f(x) \) (iii) \( \displaystyle \lim_{x \to 1^+} f(x) \) (b) Use a table of values to estimate \( \displaystyle \lim_{x \to \infty} f(x) \). (c) Use the information from parts (a) and (b) to make a rough sketch of the graph of \( f \).

Step-by-Step Solution

Answer

i) 0, ii) -∞, iii) +∞, b) +∞. Graph: Similar to a discontinuous curve with a vertical asymptote at x=1.

Step 1: Evaluate \( \lim_{x \to 0^+} f(x) \)

Substitute \( f(x) = \frac{x}{\ln x} \) and evaluate the limit as \( x \to 0^+ \) (approaching zero from the positive side). The logarithm function \( \ln x \) goes to \(-\infty\) and \( x \to 0^+ \), making \( \frac{x}{\ln x} \rightarrow 0 \). This is because the numerator approaches zero much faster than the denominator approaches negative infinity. Thus, \( \lim_{x \to 0^+} \frac{x}{\ln x} = 0 \).

Step 2: Evaluate \( \lim_{x \to 1^-} f(x) \)

Substitute \( x = 1^- \) in \( f(x) = \frac{x}{\ln x} \). As \( x \to 1^- \), \( \ln x \to 0^- \) which suggests that \( \frac{x}{\ln x} \) approaches negative infinity because the denominator is negative and very small while the numerator approaches 1. Thus, \( \lim_{x \to 1^-} \frac{x}{\ln x} = -\infty \).

Step 3: Evaluate \( \lim_{x \to 1^+} f(x) \)

Substitute \( x = 1^+ \) in \( f(x) = \frac{x}{\ln x} \). As \( x \to 1^+ \), \( \ln x \to 0^+ \) which suggests that \( \frac{x}{\ln x} \) approaches positive infinity because the denominator is positive and very small. Thus, \( \lim_{x \to 1^+} \frac{x}{\ln x} = +\infty \).

Step 4: Estimate \( \lim_{x \to \infty} f(x) \) using a table of values

To estimate \( \lim_{x \to \infty} \frac{x}{\ln x} \), consider large values such as \( x = 10, 100, 1000 \). Compute \( f(x) \):- \( f(10) = \frac{10}{\ln 10} \approx 4.34 \)- \( f(100) = \frac{100}{\ln 100} \approx 21.71 \)- \( f(1000) = \frac{1000}{\ln 1000} \approx 144.76 \)As \( x \) becomes very large, \( f(x) \) keeps increasing without bound, indicating that \( \lim_{x \to \infty} \frac{x}{\ln x} = +\infty \).

Step 5: Sketching \( f(x) \) using limit information

Use the limits from (a) and (b) to sketch the graph of \( f(x) \):- At \( x \to 0^+ \), \( f(x) \to 0 \).- At \( x \to 1^- \), \( f(x) \to -\infty \), and at \( x \to 1^+ \), \( f(x) \to +\infty \). Therefore, there is a vertical asymptote at \( x = 1 \).- As \( x \to \infty \), \( f(x) \) increases to \(+\infty\).The curve passes through the point \( (1,0) \), approaches the y-axis from the origin, dips steeply to \(-\infty\) just before \( x = 1 \) and then ascends steeply to \(+\infty\) just after \( x = 1 \). This behavior leads to a vertical asymptote at \( x=1 \) and increases without bound for large \( x \).

Key Concepts

Asymptotic Behavior

Asymptotic behavior in calculus refers to how a function behaves as it approaches a particular value or infinity. It helps us understand how a graph behaves near certain points or as the input grows very large or small. In the context of the function \(f(x) = \frac{x}{\ln x}\), we observe its asymptotic behavior at various points, such as near zero, one, and when \(x\) becomes very large. This involves evaluating limits at these critical points, showing how the function approaches a line or another function asymptotically.

• As \( x \to 0^+ \), \( f(x) \to 0 \), indicating the function value gets closer to zero.
• As \( x \to 1^- \), \( f(x) \to -\infty \) and \( x \to 1^+ \), \( f(x) \to +\infty \), revealing a drastic change around \( x = 1 \).
• As \( x \to \infty \), \( f(x) \) approaches \( +\infty \), indicating unlimited growth.

Logarithmic Functions

Logarithmic functions like \( \ln x \) play a crucial role in the function \( f(x) = \frac{x}{\ln x} \). Understanding them is essential for analyzing the behavior of the function. Logarithms have some important properties:

• The natural logarithm \( \ln x \) is only defined for positive \( x \).
• As \( x \to 0^+ \), \( \ln x \to -\infty \), conveying how rapidly they decrease on approaching zero.
• At \( x = 1 \), \( \ln 1 = 0\), making it a significant point for evaluating limits.
• As \( x \to \infty \), \( \ln x \) approaches infinity, but at a slower rate compared to linear functions like \( y = x \).

These properties influence the asymptotic behavior of \( f(x) \), making it vital to understand when evaluating limits.

Limit Evaluation

Limit evaluation is key to understanding how functions behave around particular points. We use limits to find the values that functions approach as the input values get arbitrarily close to a certain point. For \( f(x) = \frac{x}{\ln x} \), limits help us analyze behavior at \( x = 0 \), \( x = 1 \), and \( x \to \infty \).For instance, when evaluating \( \lim_{x \to 0^+} \frac{x}{\ln x} \), we observe:

• \( x \) approaches 0 while \( \ln x \) moves to \(-\infty\), forcing the entire expression \( \frac{x}{\ln x} \) to converge to 0.

For \( x \to 1 \), both sides of the limit lead to infinity, though signs differ:

• \( \lim_{x \to 1^-} f(x) = -\infty \)
• \( \lim_{x \to 1^+} f(x) = +\infty \)

Limit evaluation in each scenario reveals the function’s behavior and any discontinuities or asymptotes.

Graph Sketching

Graph sketching involves drawing a rough outline of the function based on known information, such as limits and asymptotic behavior. For \( f(x) = \frac{x}{\ln x} \), knowing how \( f(x) \) behaves at specific points and after evaluating limits is crucial. Here's what to consider:

• At \( x \to 0^+ \), graph approaches zero.
• Vertical asymptote at \( x = 1 \) as \( f(x) \) transitions from negative to positive infinity.
• For large \( x \), \( f(x) \) keeps increasing indefinitely.

Using these observations, sketch the graph showing the curve passing through key points and reflecting asymptotic tendencies. The graph dips before sharply rising post \( x = 1 \) due to its asymptotic nature.

Vertical Asymptote

A vertical asymptote occurs when a function approaches infinity as the input nears a specific value. For \( f(x) = \frac{x}{\ln x} \), we identify a vertical asymptote at \( x = 1 \). Here’s why:

• As \( x \to 1^- \), \( \ln x \to 0^- \), making \( \frac{x}{\ln x} \) go to \(-\infty\).
• As \( x \to 1^+ \), \( \ln x \to 0^+ \), making \( \frac{x}{\ln x} \) go to \(+\infty\).

This dual behavior indicates a clear boundary in the graph, where the function dramatically shifts from negative to positive infinity, resulting in a vertical asymptote at \( x = 1 \). Such asymptotes are crucial for understanding discontinuities and the overall function structure.