Problem 64 - Applied Calculus

Question

\(61-64\). Which of the two limits exists? a. \(\lim _{x \rightarrow \infty} e^{3 x}\) b. \(\lim _{x \rightarrow \infty} e^{-3 x}\)

Step-by-Step Solution

Answer

The limit \(\lim_{x \to \infty} e^{-3x}\) exists and is 0.

Step 1: Understanding the Limit Definition

We are looking at two different limits as \(x\) approaches infinity—specifically the behavior of the exponential function \(e^{3x}\) and \(e^{-3x}\).

Step 2: Analyze Limit of \(e^{3x}\)

Consider the expression \(e^{3x}\). As \(x\) approaches infinity, the exponent \(3x\) itself increases without bound. Thus, the function \(e^{3x}\) also grows exponentially towards infinity.

Step 3: Conclusion for \(e^{3x}\)

Since \(e^{3x}\) grows indefinitely as \(x\) goes to infinity, the limit \(\lim_{x \to \infty} e^{3x} = \infty\). This means the limit does not exist in the real number sense since it approaches infinity.

Step 4: Analyze Limit of \(e^{-3x}\)

Now, consider the expression \(e^{-3x}\). As \(x\) approaches infinity, the expression \(-3x\) becomes more negative. This means \(e^{-3x}\) tends to get closer and closer to zero.

Step 5: Conclusion for \(e^{-3x}\)

As \(x\) approaches infinity, \(e^{-3x}\) approaches \(0\). Therefore, \(\lim_{x \to \infty} e^{-3x} = 0\). This limit exists as a finite number.

Key Concepts

Understanding the Exponential Function

The concept of an exponential function is central to calculus and mathematical analysis. In its simplest form, the exponential function is written as \(e^x\), where \(e\) is the base of the natural logarithm, approximately equal to 2.71828. It's an essential function as it describes a constant rate of growth or decay. Here, the functions we are analyzing are \(e^{3x}\) and \(e^{-3x}\).

- In an exponential function like \(e^{3x}\), when the variable \(x\) increases, the exponent causes the entire expression to grow larger. It describes exponential growth.
- Conversely, in \(e^{-3x}\), as \(x\) increases, the negative exponent causes the value of the function to diminish, indicating exponential decay.

These properties make exponential functions crucial for modeling real-world phenomena like population growth, radioactive decay, and interest calculations.

The Concept of Infinity in Calculus

Infinity is a concept that we use to explain limits and behaviors of functions as they grow unbounded. In calculus, we often examine what happens to functions as they "approach" infinity. This doesn't imply a number that can be reached, but rather describes a function's behavior trend.

For example, when we analyze \(\lim_{x \to \infty} e^{3x}\), infinity captures the idea that as \(x\) gets larger and larger, \(e^{3x}\) increases without any stopping point. It essentially grows exponentially with no bound.
In the case of \(e^{-3x}\), as \(x\) increases towards infinity, the function itself approaches zero—a finite number. Here, infinity describes a trajectory where \(e^{-3x}\) tends towards zero but never actually becomes it.

Thus, infinity helps us understand whether a limit exists or how a function behaves towards its extremities.

Performing Limit Analysis

Limit analysis is a fundamental tool in calculus that examines the behavior of functions as they approach certain points or conditions, such as infinity. By assessing limits, we can determine if a function stabilizes to a specific value or continues growing without bound.

- With \(\lim_{x \to \infty} e^{3x}\), we find that the function doesn't stabilize at a particular number. Instead, it continues growing indefinitely. Hence, we say its limit does not "exist" in the finite sense because it evaluates to infinity.
- However, for \(\lim_{x \to \infty} e^{-3x}\), the function approaches 0 in a measurable and stable manner. This indicates that the limit does exist at a finite value.

These analyses are crucial for exploring the boundaries of functions and understanding behaviors in real-world contexts, such as predicting how quickly a substance dissolves or how investments grow over time.