Problem 35 - Calculus with Concepts in Calculus

Question

Find the horizontal asymptote of the graph of the function. Then sketch the graph of the function. $$ f(x)=\ln \left(1+e^{x}\right) $$

Step-by-Step Solution

Answer

The horizontal asymptote is \( y = 0 \).

Step 1: Understanding Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the input values become arbitrarily large (positive or negative). For a function \( f(x) \), if \( \lim_{{x \to \infty}} f(x) = L \) or \( \lim_{{x \to -\infty}} f(x) = L \), then \( y = L \) is a horizontal asymptote of the function.

Step 2: Analyze Behavior as \( x \to \infty \)

Consider the function \( f(x) = \ln(1+e^x) \). As \( x \to \infty \), \( e^x \) becomes very large, so \( 1+e^x \) approaches \( e^x \). Therefore, \( f(x) \approx \ln(e^x) = x \). Hence, \( \lim_{{x \to \infty}} f(x) = \infty \). This indicates that there is no horizontal asymptote as \( x \to \infty \).

Step 3: Analyze Behavior as \( x \to -\infty \)

For \( x \to -\infty \), \( e^x \) approaches zero, making \( 1+e^x \approx 1 \). Hence, \( f(x) \approx \ln(1) = 0 \). This implies that \( \lim_{{x \to -\infty}} f(x) = 0 \). Thus, there is a horizontal asymptote \( y = 0 \) as \( x \to -\infty \).

Step 4: Sketching the Graph

For \( x \to -\infty \), the graph approaches the horizontal asymptote \( y = 0 \). As \( x \to \infty \), the graph increases without bound. Therefore, the graph starts at the horizontal asymptote on the left and rises continuously to the right. The graph should smoothly curve upwards, reflecting the behavior of the natural logarithm as it relates to \( e^x \).

Key Concepts

Limits

Limits are fundamental in understanding the behavior of functions as they approach specific values. A limit describes what a function does as the input gets closer to a particular number. It tells us where a function is "heading," even if it never actually reaches that point.

When dealing with horizontal asymptotes and limits, we are often interested in the behavior of a function as the input values become extremely large (approaching infinity) or extremely small (approaching negative infinity). A horizontal asymptote is a line that the graph of a function approaches as the input heads towards infinity or negative infinity. Some key points about limits include:

• If \( \lim_{{x \to \infty}} f(x) = L \), then \( y = L \) is a horizontal asymptote for the function to the right.
• If \( \lim_{{x \to -\infty}} f(x) = L \), then \( y = L \) is a horizontal asymptote to the left.
• Understanding limits allows us to predict and sketch the asymptotic behavior of functions even beyond their clear domain.

Natural Logarithm

The natural logarithm, written as \( \ln(x) \), is a logarithm with a special base known as Euler's number (approximately 2.718). It is one of the most used logarithms in mathematics due to its natural properties, especially when dealing with continuous growth or decay processes.

For the function \( f(x) = \ln(1+e^x) \), as \( x \to -\infty \), the expression \( e^x \) approaches zero. This transformation simplifies the function to \( \ln(1) = 0 \). This tells us that as \( x \to -\infty \), the function approaches zero, which is why \( y = 0 \) is a horizontal asymptote.

The natural logarithm has some unique properties:

• It is the inverse of the exponential function, meaning \( \ln(e^x) = x \).
• \( \ln(1) = 0 \), reflecting its base's properties succinctly.
• Natural logarithms relate to exponential growth in fields like biology and physics.

Exponential Functions

Exponential functions involve a constant base raised to a variable exponent, often expressed as \( e^x \). The number \( e \), known as Euler's number, approximately equals 2.718 and is important for describing growth processes.

In the function \( f(x) = \ln(1+e^x) \), the exponential function \( e^x \) plays a crucial role in determining the behavior and asymptotes of the function. Here's how it works:

• As \( x \to \infty \), \( e^x \) becomes very large, which makes \( 1+e^x \) approximate \( e^x \). Hence, \( \ln(1+e^x) \approx \ln(e^x) = x \). The function grows without bound, meaning it doesn't approach any particular horizontal line.
• Conversely, as \( x \to -\infty \), \( e^x \) approaches zero, making \( 1+e^x \approx 1 \), and thus \( \ln(1) = 0 \).
• Exponential growth described by \( e^x \) is faster compared to polynomial growth, often leading to significant changes in the behavior of functions involving \( e^x \).
• Exponential functions are not just theoretical; they model real-world phenomena such as population growth and radioactive decay.