Problem 11 - Applied Calculus

Question

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{a \rightarrow-\infty} e^{2 a} $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Understand the Limit Expression

We're tasked with evaluating \( \lim_{a \to -\infty} e^{2a} \). This means we need to determine the behavior of the expression \( e^{2a} \) as \( a \) approaches negative infinity.

Step 2: Analyze the Function Behavior

The function \( e^x \) approaches zero as \( x \) becomes very negative. Since \( 2a \) will become negatively large as \( a \to -\infty \), we need to understand that \( e^{2a} \) will also approach zero when \( a \to -\infty \).

Step 3: Write the Conclusion

From the analysis, since \( e^{2a} \) approaches zero as \( a \to -\infty \) because exponents become negative and exponents of \( e \) diminish to a value close to zero, the limit is zero.

Key Concepts

Exponential Functions

Exponential functions play a crucial role in mathematics, especially for describing growth and decay processes. The general form of an exponential function is \( f(x) = a \cdot e^{bx} \), where \( e \) is a constant approximately equal to 2.71828, known as the base of natural logarithms. In this context, \( e^{2a} \) is an example of an exponential function.

• The base \( e \) makes these functions unique because of their continuous rate of growth.
• When the exponent \( 2a \) becomes very large or very small, it dictates the behavior of the entire function.

For instance, when an exponential function has a negative exponent, such as \( e^{-x} \), it represents exponential decay. As the value of \( x \) increases, the value of \( e^{-x} \) decreases swiftly towards zero. This concept of exponential decay is important for understanding how functions like \( e^{2a} \) behave as \( a \) approaches negative infinity.

Behavior at Infinity

Understanding the behavior of mathematical functions at infinity helps predict what happens as values become extremely large or small. When considering limits, we often analyze how a function behaves as its input either increases or decreases without bound.

• For functions such as \( e^{2a} \), analyzing \( a \) approaching negative infinity provides insight into the function's decreasing pattern.
• As \( a \rightarrow -\infty \), the term \( 2a \) also tends towards negative infinity.

This means the exponential function \( e^{2a} \) approaches zero. This behavior is a key feature of many limits, showing how functions can trend towards a value, often zero, as their inputs grow infinitely large or small. Recognizing these trends is essential for effectively evaluating limits.

Limit Evaluation

Limit evaluation is a fundamental concept in calculus used to understand how functions behave as their variables approach specific values. To evaluate the limit \( \lim_{a \to -\infty} e^{2a} \), several steps are involved:

• Identify what the expression looks like as the variable approaches infinity or negative infinity.
• Understand how the function behaves, using proven mathematical behaviors like exponential growth or decay.

Within this framework, as \( a \to -\infty \), the term \( 2a \) decreases without bound. Consequently, \( e^{2a} \) diminishes towards zero due to the properties of exponential functions with negative exponents.
Evaluating limits isn't just about finding a value—it also involves understanding the underlying function behavior. Recognizing relevant behaviors ensures that we reach the correct conclusions and enhances your ability to tackle more complex calculus problems.