Problem 46 - Calculus Early Transcendentals

Question

Determine the end behavior of the following transcendental functions by analyzing appropriate limits. Then provide a simple sketch of the associated graph, showing asymptotes if they exist. $$f(x)=2^{x}$$

Step-by-Step Solution

Answer

Based on the step-by-step solution provided, answer the following prompt: Question: Describe the end behavior of the function \(f(x) = 2^x\). Answer: As \(x \to \infty\), the function \(f(x) = 2^x\) increases without bound, and as \(x \to -\infty\), the function approaches 0, with a horizontal asymptote at y=0.

Step 1: Determine the limit as x approaches positive infinity

To find the limit as x approaches positive infinity, we will evaluate the limit as \(x \to \infty\): $$\lim_{x \to \infty} 2^x$$ As x becomes very large, the value of \(2^x\) also becomes very large. Therefore, the limit as x approaches positive infinity is: $$\lim_{x \to \infty} 2^x=\infty$$

Step 2: Determine the limit as x approaches negative infinity

Now, we need to evaluate the limit as \(x\to -\infty\): $$\lim_{x \to -\infty} 2^x$$ As x becomes very negative, the value of \(2^x\) approaches 0. Therefore, the limit as x approaches negative infinity is: $$\lim_{x \to -\infty} 2^x=0$$

Step 3: Identify asymptotes

As we found in Step 2, the function approaches 0 as \(x \to -\infty\). This indicates the existence of a horizontal asymptote at y=0.

Step 4: Sketch the graph

Using the information from Steps 1-3, we will now sketch the graph of the function \(f(x)=2^x\). 1. As \(x\to \infty\), the function increases without bound (grows very large). 2. As \(x\to -\infty\), the function approaches 0. 3. There is a horizontal asymptote at y=0 as x approaches negative infinity. From these observations, we can sketch the graph of \(f(x)=2^x\) with an increasing curve that approaches the y-axis but never touches it, and a horizontal asymptote at y=0 as the function approaches negative infinity.

Key Concepts

Transcendental Functions

Transcendental functions go beyond algebraic functions, which involve only basic operations such as addition, subtraction, multiplication, division, and taking roots. They involve more complex operations like exponentiation and trigonometric functions. A famous example is the exponential function, such as \(f(x) = 2^x\), which grows dramatically different from polynomial functions.

Transcendental functions often appear when modeling real-world phenomena like population growth, radioactive decay, and compound interest, providing essential tools for understanding complex systems.

Learning about these functions can help us understand their unique characteristics, such as non-algebraic growth patterns and their ability to pass beyond the confines of polynomial expressions. Recognizing the end behavior of transcendental functions helps in creating accurate predictions and understanding various natural and scientific processes.

Limits and Continuity

Understanding limits is crucial for determining the end behavior of functions. A limit describes how a function behaves as the variable approaches a particular value. In the context of transcendental functions like \(f(x) = 2^x\), examining limits at both positive and negative infinity reveals how the function behaves as \(x\) becomes exceedingly large or very negative.

One important aspect is continuity, which tells us that between any two points on a real line, there's a smooth, connected path. For \(f(x) = 2^x\), the function is continuous across all real numbers, meaning there are no breaks or jumps. This continuous nature allows us to easily predict how the function behaves for various values of \(x\).

• If \(x \to \infty\), then \(2^x \to \infty\), indicating endless growth.
• If \(x \to -\infty\), then \(2^x \to 0\), showing that while the function decreases, it never truly reaches zero.

Thus, exploring limits and ensuring continuity are vital for correctly understanding the function's behavior at its endpoints.

Asymptotes

An asymptote is a line that a graph approaches but never touches. In mathematical functions, asymptotes can be horizontal, vertical, or oblique. For the function \(f(x) = 2^x\), the concept of a horizontal asymptote is critical.

Horizontal asymptotes reveal the behavior of a function as \(x\) approaches infinity in either direction. For \(f(x) = 2^x\), at \(x \to -\infty\), the value of \(f(x)\) approaches 0, so \(y = 0\) acts as a horizontal asymptote.

This asymptotic behavior is particularly evident in exponential functions where the output becomes infinitely close to a particular value without actually reaching it. Understanding the placement of asymptotes helps sketch graphs accurately, indicating regions where functions change sharply or level out.

• Horizontal Asymptote: For \(f(x) = 2^x\), \(y=0\) when \(x \to -\infty\).

Recognizing asymptotic behavior and identifying these invisible boundaries provide key insights into the graphical representation and long-term trends of functions.