Problem 22 - Precalculus: Mathematical for Calculus

Question

Use a graphing device to determine whether the limit exists. If the limit exists, estimate its value to two decimal places. $$\lim _{x \rightarrow 0} \frac{1}{1+e^{1 / x}}$$

Step-by-Step Solution

Answer

The limit does not exist.

Step 1: Understand the Limit Problem

We are asked to find the limit as \( x \) approaches 0 for the function \( \frac{1}{1+e^{1/x}} \). To solve this, we need to consider the behavior of the function as \( x \) approaches 0 from both the left (negative values of \( x \)) and the right (positive values of \( x \)).

Step 2: Analyze the Function Behavior as x Approaches 0 from the Left

As \( x \to 0^- \), the exponent \( \frac{1}{x} \to -\infty \), making \( e^{1/x} \to 0 \). Therefore, the expression \( 1 + e^{1/x} \) approaches 1, and hence \( \frac{1}{1+e^{1/x}} \to 1 \).

Step 3: Analyze the Function Behavior as x Approaches 0 from the Right

As \( x \to 0^+ \), the exponent \( \frac{1}{x} \to +\infty \), making \( e^{1/x} \to \infty \). Therefore, \( 1 + e^{1/x} \) approaches infinity, and the fraction \( \frac{1}{1+e^{1/x}} \to 0 \).

Step 4: Conclude Whether the Limit Exists

Since the left-hand limit (as \( x \) approaches 0 from the left) equals 1, and the right-hand limit (as \( x \) approaches 0 from the right) equals 0, these two values are not equal. Therefore, the limit does not exist.

Key Concepts

Graphical Analysis

In calculus, graphical analysis is a key tool that helps in understanding the behavior of functions visually. By plotting a function, one can observe how it behaves as it approaches certain values. It offers a visual insight into limits, continuity, and differentiability. When examining limits, graphical analysis allows students to see:

• Where the function is heading as the input gets closer to a certain value.
• If the function has any discontinuities or asymptotic behavior.

In the case of the limit problem given, using a graphing device can clearly show the differing behaviors as the variable approaches zero from both sides. It can provide a visible confirmation that the function does not approach the same value as it nears zero from the left and right. Thus, it visually supports the mathematical conclusion that the limit does not exist as the values differ between approaching from the left and the right.

One-Sided Limits

One-sided limits are a fundamental concept in calculus. They focus on the value that a function approaches as the input approaches a particular point from one specific side.For two types of one-sided limits, we have:

• Left-hand limit: Denoted as \( \lim_{x \to a^-} f(x) \), it examines the function as it approaches \( a \) from the left (negative direction).
• Right-hand limit: Denoted as \( \lim_{x \to a^+} f(x) \), it focuses on the approach from the right (positive direction).

In our problem, as \( x \to 0^- \), the behavior of \( \frac{1}{1+e^{1/x}} \) trends towards 1, as \( e^{1/x} \) becomes negligible compared to 1. From the right, \( x \to 0^+ \), it trends towards 0, due to \( e^{1/x} \) increasing rapidly. Because these one-sided limits do not agree, the two-sided limit does not exist. Understanding one-sided limits helps explain why the limit result differs when approached from different directions.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions show rapid growth or decay rates, depending on the sign of the exponent.Some key properties of exponential functions include:

• The function \( e^x \) where \( e \) is approximately 2.71828, is the base of natural logarithms.
• Exponential growth occurs for positive exponents, and decay for negative exponents.

In our exercise, the component \( e^{1/x} \) is an exponential function where the exponent changes dramatically as \( x \) approaches zero. This expression dramatically affects the behavior of the entire function \( \frac{1}{1+e^{1/x}} \), showing extreme changes from negligible to massive values, depending on the sign and magnitude of the variable extract. Thus, understanding the properties of exponential functions provides insight into how non-linear responses can affect limit behavior.