Problem 8 - Calculus Early Transcendentals

Question

Use a sketch to find the end behavior of \(f(x)=\ln x\)

Step-by-Step Solution

Answer

Answer: The end behavior of the function \(f(x) = \ln x\) is as follows: - As \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\) - As \(x \rightarrow 0^+\), \(f(x) \rightarrow -\infty\)

Step 1: Sketch the graph of \(\ln x\)

Plot the function \(f(x) = \ln x\) on the xy-plane. You can make use of a graphing calculator, online tools like Desmos or WolframAlpha, or analyze the function yourself to sketch it. The graph of \(\ln x\) passes through the point (1,0) and increases towards positive infinity as x increases, while it approaches negative infinity as x approaches 0 from the positive side.

Step 2: Determine the end behavior as \(x\rightarrow\infty\)

Analyze how the function behaves as x approaches positive infinity. As \(x\) gets larger, the natural logarithm of \(x\) also gets larger. Thus, we can say that as \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\).

Step 3: Determine end behavior as \(x\rightarrow0^+\)

Now, analyze the behavior of the function as \(x\) approaches 0 from the right side (positive side). As \(x\) gets closer to 0, the natural logarithm of \(x\) goes to negative infinity. Thus, we can say that as \(x \rightarrow 0^+\), \(f(x) \rightarrow -\infty\).

Step 4: Summarize the end behavior of \(f(x) = \ln x\)

The end behavior of the function \(f(x) = \ln x\) is as follows: - As \(x \rightarrow \infty\), \(f(x) \rightarrow \infty\) - As \(x \rightarrow 0^+\), \(f(x) \rightarrow -\infty\) Now, you have successfully determined the end behavior of the given function by sketching and analyzing the graph of \(\ln x\).

Key Concepts

Logarithmic Function

A logarithmic function is a type of mathematical function that represents the logarithm of a number. The most common logarithmic function is the natural logarithm, denoted as \(f(x) = \ln x\).
This function uses Euler's number \(e\) as the base, where \(e\) is approximately equal to 2.718. In simpler terms, a logarithm answers the question: "To what exponent must the base \(e\) be raised, to produce a certain number?"
For instance, \(\ln e = 1\) because \(e^1 = e\). Logarithmic functions are unique in that:

• They are only defined for positive values of \(x\).
• They often arise in the context of decay processes, such as radioactive decay, and in mathematical models like compound interest calculations.

Understanding the properties of logarithmic functions is key to sketching their graphs accurately.

Graph Sketching

Graph sketching is the art of drawing a function on a coordinate plane to analyze its behavior and characteristics visually. When sketching the graph of \( \ln x \), consider these essential points:

• The graph passes through the point \((1, 0)\), serving as a fundamental reference.
• The function is undefined for \( x \leq 0 \), which means the graph only exists in the positive x-axis region.
• As \(x\) increases, the function \( \ln x \) also increases, but at a decreasing rate.

The graph of \( \ln x \) slopes gently upwards to the right, showing a slow growth towards infinity.
It's also crucial to note that as \(x\) approaches 0 from the positive side, the curve dives sharply towards negative infinity, illustrating its infinite descent as it nears the vertical asymptote at the y-axis.

Limits

Limits in mathematics help us analyze how a function behaves as the input approaches a particular value, usually towards infinity. When discussing \( \ln x \), understanding limits provides a clearer picture of its end behavior.
In the graph of \( \ln x \):

• As \(x \to \infty\), \(\ln x \to \infty\) too; the function grows without bounds.
• As \(x \to 0^+\), \(\ln x \to -\infty\); the function decreases infinitely as it nears the y-axis.

This behavior is critical because it highlights the non-linear nature of logarithmic growth and decay, contrasting significantly with linear functions.
Practically, it tells us, for instance, that while \( \ln x \) grows indefinitely, it does so at an ever-decreasing rate, reflecting nuances in patterns like population growth or signal attenuation.

Infinity Behavior

Infinity behavior describes what happens to a function's output as the input grows larger and larger, either positively or negatively. For \( \ln x \), this concept is quite illuminating:

• When \( x \rightarrow \infty \), the function \( \ln x \) heads toward infinity. It reveals continuous growth, but gradually slowing as values rise.
• Conversely, as \( x \rightarrow 0^+ \), the function plunges towards negative infinity, showing a dramatic descent approaching the y-axis. This represents a vertical asymptote, where values of \( f(x) \) decrease without bound.

Understanding this behavior equips us with insights into real-world applications, such as understanding algorithms in computer science where input size impacts outcomes, or physics scenarios with exponentially diminishing resources.
Ultimately, infinity behavior analysis helps depict a function's distant future and past, extending beyond simple numerical calculations.