Problem 33 - Calculus Early Transcendentals

Question

15-36 Find the limit. $$ \lim _{x \rightarrow \infty} \frac{1-e^{x}}{1+2 e^{x}} $$

Step-by-Step Solution

Answer

The limit is \( \frac{-1}{2} \).

Step 1: Understand the Problem

We need to find the limit of the function \( \frac{1-e^{x}}{1+2e^{x}} \) as \( x \rightarrow \infty \). This means we want to know what value the function approaches as \( x \) becomes very large.

Step 2: Examine the Behavior of \( e^{x} \)

Recall that \( e^{x} \) grows very rapidly to \( \infty \) as \( x \rightarrow \infty \). Therefore, terms involving \( e^{x} \) in the numerator and denominator will dominate the other constant terms (1 in this case) as \( x \) becomes very large.

Step 3: Simplify the Expression by Dividing by \( e^x \)

To simplify, divide both the numerator and the denominator by \( e^x \):\[\lim _{x ightarrow \infty} \frac{1-e^{x}}{1+2e^{x}} = \lim _{x \rightarrow \infty} \frac{\frac{1}{e^x}-1}{\frac{1}{e^x}+2}\]As \( x \rightarrow \infty \), \( \frac{1}{e^x} \rightarrow 0 \).

Step 4: Evaluate the Limit

Substitute \( \frac{1}{e^x} = 0 \) as \( x \rightarrow \infty \):\[\lim _{x \rightarrow \infty} \frac{0 - 1}{0 + 2} = \frac{-1}{2}\]Thus, the limit is \( \frac{-1}{2} \).

Key Concepts

Exponential Functions

An exponential function is a mathematical expression in the form of \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. Unlike polynomial functions, exponential functions grow rapidly as the value of \( x \) increases. This rapid growth is significant in problems that involve limits, especially as \( x \) approaches infinity. For positive values of \( x \), \( e^x \) becomes very large, overtaking any constant terms in the numerator or denominator of a fraction. This behavior essentially dictates how the function behaves at infinity and impacts the solution for limit problems. The dominance of exponential terms over constant terms is a pattern often leveraged to simplify expressions resulting in more manageable forms when calculating limits.

Infinity Concept

The concept of infinity is crucial in calculus, especially when dealing with limits. The notation \( x \rightarrow \infty \) describes the behavior of a function as \( x \) becomes indefinitely large. Here, infinity is not a number to be reached, but an idea representing the unbounded growth of \( x \). In the given problem \( \frac{1-e^{x}}{1+2e^{x}} \), as \( x \) grows, the terms \( e^x \) in both the numerator and denominator influence the outcome of the limit. Recognizing the infinity concept allows us to evaluate the behavior of mathematical expressions. Specifically, terms like \( \frac{1}{e^x} \) approach zero as \( x \rightarrow \infty \), simplifying complex fractions significantly when calculating limits.

Asymptotic Behavior

Asymptotic behavior refers to the manner in which a function behaves as it approaches a particular line or curve, commonly as the variable approaches infinity in limit problems. In the exercise \( \frac{1-e^{x}}{1+2e^{x}} \), as \( x \) becomes very large, the term \( e^x \) grows, causing any constant parts of the fraction to have negligible influence on the limit. By simplifying the fraction with respect to \( \frac{e^x} \), the asymptotic behavior dictates that constants like 1 become insignificant compared to \( e^x \). Therefore, the asymptotic nature significantly affects the outcome of limits involving exponential functions. This concept helps us achieve solutions like \( \frac{-1}{2} \) for the given problem, showing how significant terms overshadow smaller, constant terms as \( x \rightarrow \infty \).