Problem 24 - Contemporary Precalculus

Question

Use the Big-Little Principle to explain why \(e^{x}+e^{-x}\) is approximately equal to \(e^{x}\) when \(x\) is large.

Step-by-Step Solution

Answer

Question: Explain why \(e^{x} + e^{-x}\) is approximately equal to \(e^{x}\) when \(x\) is large using the Big-Little Principle. Answer: When x is large, the value of \(e^{-x}\) becomes negligible compared to \(e^x\), according to the Big-Little Principle. This is because the ratio \(|e^{-x}|/|e^x|\) approaches 0 as x approaches infinity. Therefore, for large values of x, \(e^{x} + e^{-x}\) is approximately equal to \(e^x\).

Step 1: Expression for Big-Little Principle

The Big-Little Principle states that if \(f(x)\) and \(g(x)\) are two functions on the same variable \(x\) and if \(|f(x)|/|g(x)| \to 0\) as \(x \to \infty\), then \(f(x)\) is negligible in comparison to \(g(x)\).

Step 2: Define the functions for our problem

In our exercise, let's set \(f(x) = e^{-x}\) and \(g(x) = e^x\). Now, we want to analyze the behavior of these functions as x becomes large and find the ratio \(|f(x)|/|g(x)|\).

Step 3: Calculate the ratio of the functions

To analyze the behavior of the functions, we need to compute the following ratio: $$\lim_{x \to \infty} \frac{|e^{-x}|}{|e^x|}$$

Step 4: Simplify the ratio and find the limit

We can simplify the ratio as: $$\lim_{x \to \infty} \frac{e^{-x}}{e^x}$$ Dividing by exponential expressions with the same base means subtracting their exponents: $$\lim_{x \to \infty} e^{-x-x}$$ $$\lim_{x \to \infty} e^{-2x}$$ Now, when \(x\) is large, the value of \(-2x\) becomes increasingly negative and the exponential value approaches 0. Therefore, the limit is: $$\lim_{x \to \infty} e^{-2x} = 0$$

Step 5: Use Big-Little Principle

According to the Big-Little Principle, since the limit of the ratio \(|f(x)|/|g(x)|\) is 0 as x approaches infinity, we conclude that \(f(x) = e^{-x}\) is negligible compared to \(g(x) = e^x\) when x is large. So, \(e^x + e^{-x}\) is approximately equal to \(e^x\) for large values of x.

Key Concepts

Exponential Functions

Exponential functions are a key type of mathematical function characterized by a base raised to a power that is a variable. The most common form is expressed as \(f(x) = a^x\), where \(a\) is a constant and \(x\) is the exponent. A special and frequently used exponential function is \(f(x) = e^x\), where \(e\) represents Euler's number, approximately equal to 2.71828.
Exponential functions are unique because they involve constant multiplicative rates of change, meaning as \(x\) increases or decreases, the function's value changes by a consistent factor. This property results in very rapid growth or decay. Key characteristics of exponential functions include:

• Rapid growth or decay, depending on the base.
• Non-zero values only, implying no real roots.
• Curves that are always increasing or decreasing; never both.

An appreciation of these functions is essential for understanding complex phenomena in fields like biology, economics, and physics, where exponential growth or decay rates are common.

Function Behavior as x Approaches Infinity

When analyzing functions, understanding their behavior as \(x\) approaches infinity is critical. This process often involves evaluating limits, which helps classify functions as either dominating or negligible.
For the exercise, we explore the Big-Little Principle, which assists in identifying dominating components in a function as \(x\) gets large. Let's take the example of two exponential functions: \(e^x\) and \(e^{-x}\). Here are the steps involved in analyzing their behavior as \(x\) tends to infinity:

• Consider the limit \(\lim_{x \to \infty} \frac{|e^{-x}|}{|e^x|}\), where you check how one function's magnitude compares with another's as \(x\) increases.
• Find the limit: \(\lim_{x \to \infty} e^{-2x} = 0\).
• Conclude, by the Big-Little Principle, \(e^{-x}\) is negligible compared to \(e^x\) for large \(x\).

Recognizing this behavior is useful in simplifying functions and solving complex mathematical problems by focusing on the most significant terms as \(x\) grows without bound.

Approximation in Mathematics

Approximation is a powerful tool in mathematics allowing for simplification when solving problems, especially when exact solutions are unnecessary or impractical. In high-level mathematics, approximating functions often involves deciding which terms can be ignored for ease of calculation without significantly affecting the answer.
The given exercise serves as an excellent illustration. By acknowledging \(e^{-x}\) as negligible compared to \(e^x\), we approximate \(e^x + e^{-x} \approx e^x\) when \(x\) is large. This approximation provides a simplified model that captures the core behavior of the function without unnecessary complexity.
Approximations are common in:

• Breaking down complex expressions.
• Focusing on dominant terms for large or small values of variables.
• Enhancing computational efficiency and clarity.

However, it's crucial to carefully consider the conditions under which approximations are valid, as overlooking relevant terms might lead to significant errors in certain contexts.