Problem 95 - Calculus: Early Transcendentals

Question

Find the horizontal asymptotes of each function using limits at infinity. $$f(x)=\frac{2 e^{x}+3}{e^{x}+1}$$

Step-by-Step Solution

Answer

Answer: The horizontal asymptote of the given function is y = 2.

Step 1: Find the limit as x approaches positive infinity

To find the limit as x approaches positive infinity, we need to find: $$\lim_{x\to\infty} \frac{2 e^{x}+3}{e^{x}+1}$$ Since both the numerator and the denominator have e^x terms, we can divide both the numerator and the denominator by e^x. $$\lim_{x\to\infty} \frac{2 e^{x}+3}{e^{x}+1} = \lim_{x\to\infty} \frac{2+\frac{3}{e^{x}}}{1+\frac{1}{e^{x}}}$$ As x approaches positive infinity, the terms with e^x in the denominator will approach 0: $$\lim_{x\to\infty} \frac{2+\frac{3}{e^{x}}}{1+\frac{1}{e^{x}}} = \frac{2+0}{1+0} = 2$$ So the graph of the function approaches the line y = 2 as x approaches positive infinity, and this is one horizontal asymptote.

Step 2: Find the limit as x approaches negative infinity

To find the limit as x approaches negative infinity, we need to find: $$\lim_{x\to-\infty} \frac{2 e^{x}+3}{e^{x}+1}$$ Using the same technique from Step 1, we divide both the numerator and the denominator by e^x: $$\lim_{x\to-\infty} \frac{2 e^{x}+3}{e^{x}+1} = \lim_{x\to-\infty} \frac{2+\frac{3}{e^{x}}}{1+\frac{1}{e^{x}}}$$ As x approaches negative infinity, the terms with e^x in the denominator will approach infinity: $$\lim_{x\to-\infty} \frac{2+\frac{3}{e^{x}}}{1+\frac{1}{e^{x}}} = \frac{2+0}{1+0} = 2$$ So the graph of the function approaches the line y = 2 as x approaches negative infinity as well.

Step 3: Identify the horizontal asymptotes

Since the graph of the function approaches the line y = 2 as x approaches both positive and negative infinity, the horizontal asymptote is at y = 2. The horizontal asymptote of the given function is y = 2.

Key Concepts

Limits at Infinity

When analyzing the limits of a function as the variable approaches infinity, you essentially investigate the behavior of the function further along the x-axis, both positively and negatively. The core idea is to observe what value the function gets closer to, even if it never quite reaches. This value, if it is a constant number, will often define the horizontal asymptote of the function.
For example, taking the function given:

• For the limit as \( x \) approaches positive infinity, both the numerator \( 2e^x + 3 \) and the denominator \( e^x + 1 \) have the term \( e^x \), which increases rapidly.
• By dividing every term in the numerator and the denominator by \( e^x \), the expression simplifies.
• As \( x \to \infty \), terms like \( \frac{3}{e^x} \) tend towards zero, simplifying the fraction to \( \frac{2}{1} = 2 \). This results in a horizontal asymptote of \( y = 2 \).
• Similarly, as \( x \to -\infty \), we still find the graph approaching \( y = 2 \) because of the behavior of \( e^x \) in the negative direction, still leading to terms diminishing to zero.

Exponential Functions

Exponential functions are powerful mathematical expressions where the independent variable, usually represented as \( x \), is located in the exponent. These functions are characterized by broad behaviors including rapid growth or decay depending on the base of the exponent. Typically, when examining limits involving exponential functions at infinity, it is crucial to note how they can dominate other terms in a function, substantially influencing horizontal asymptotes.

• In the given function \( f(x)=\frac{2 e^{x}+3}{e^{x}+1} \), the \( e^x \) term grows exponentially as \( x \to \infty \).
• Exponential terms are particularly significant as they can dwarf constant terms like \( 3 \) and \( 1 \), simplifying the analysis to focus on the functions of \( e^x \) over large values of \( x \).
• Asymptotic analysis becomes manageable by leveraging the rapid growth (or decay) of \( e^x \), allowing us to see the dominating behavior, especially as \( x \to \pm \infty \).

Graphing Rational Functions

Rational functions are expressions that can be written as the ratio of two polynomials. Graphing these types of functions involves discerning critical features like intercepts, holes, and asymptotes, which include vertical and horizontal asymptotes. Horizontal asymptotes are closely linked to the behavior at infinity, providing a baseline the graph approaches but does not necessarily touch.
Let's elaborate further:

• For the function \( f(x)=\frac{2 e^{x}+3}{e^{x}+1} \), consider it as a ratio of exponential functions.
• The horizontal asymptote provides a clear picture of how the function behaves for very large (both positive and negative) values of \( x \).
• Graphically, as \( x \) approaches infinity in either direction, observe how the curve flattens toward the asymptote, in this case, \( y = 2 \). This visual check confirms our algebraic analysis.

Understanding these characteristics assists students in predicting the graph's behavior and ensuring accuracy in plotting the function over a range of \( x \) values.