Problem 13 - 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 \( \infty \).

Step 1: Understand the Expression

We are given the limit expression \( \lim_{a \rightarrow -\infty} e^{-2a} \). This involves the function \( e^{-2a} \), where \( a \) approaches negative infinity.

Step 2: Rewrite the Exponential Term

Note that the expression \( e^{-2a} \) can be rewritten as \( \frac{1}{e^{2a}} \). This is because \( e^{-x} = \frac{1}{e^x} \).

Step 3: Evaluate the Limit of Exponent with Negative Infinity

As \( a \rightarrow -\infty \), the term \( 2a \) also approaches \(-\infty\). This means \( e^{2a} \rightarrow e^{-\infty} \), which equals \( 0 \) because the exponential function approaches zero as the exponent becomes very large negative.

Step 4: Determine the Limit of the Expression

Since \( e^{2a} \rightarrow 0 \), it follows that \( \frac{1}{e^{2a}} \) becomes a division by a number approaching zero, heading towards infinity. Thus, \( \lim_{a \rightarrow -\infty} e^{-2a} = \infty \).

Key Concepts

Exponential Functions

Exponential functions are an essential part of calculus, especially when evaluating limits. These functions are defined by an expression in the form of \( y = e^x \), where \( e \) represents the base of the natural logarithm, approximately equal to 2.71828.
This base, \( e \), is unique because it arises naturally in various mathematical processes, especially in those involving growth and decay.
In an exponential function, changes in the exponent lead to rapid changes in the function's value.

• Exponential growth means that as the exponent increases, the function value skyrockets.
• Exponential decay denotes the function value plummeting as the exponent decreases.

For the function in our exercise \( e^{-2a} \), the negative sign in the exponent indicates decay.
As \( a \) becomes more negative, it suggests significant decay, reducing the value of the exponential function quickly towards zero.

Infinity Concepts

Infinity is not a number but a concept that describes something without bound or end. In calculus, we often deal with limits approaching infinity, denoted by \( \infty \) or \( -\infty \).
Understanding these can clarify how functions behave as they stretch towards extremely large or small values.

• Positive infinity (\( \infty \)) indicates a function trending upwards without limit.
• Negative infinity (\( -\infty \)) indicates a function trending downwards without an end.

In our exercise, when we evaluate the limit \( \lim _{a \rightarrow -\infty} \), we're exploring what happens as \( a \) becomes very large negatively.
This helps us determine the behavior of functions like \( e^{-2a} \), forecasting that as \( a \) stretches toward \( -\infty \), the function trends dramatically towards positive infinity.
Recognizing these infinity concepts aids in evaluating limits accurately.

Limit Evaluation

Evaluating limits involves analyzing how a function behaves as the input comes arbitrarily close to a particular value. Limits are fundamental to understanding continuous change in calculus.
To evaluate a limit, consider what happens to the function values as the input approaches either the specified point or infinity.

• Direct substitution often works for real number limits.
• Algebraic manipulation, like factoring or rationalizing, can also be helpful.
• L'Hôpital's rule may be used if the function involves indeterminate forms.

In the problem \( \lim_{a \rightarrow -\infty} e^{-2a} \), evaluating the limit includes changing the exponential expression to \( \frac{1}{e^{2a}} \).
Understanding that \( e^{2a} \rightarrow 0 \) as \( a \rightarrow -\infty \) helps us see that \( \frac{1}{0} \) trends towards infinity.
Thus, the limit evaluation in this scenario concludes with \( \lim_{a \rightarrow -\infty} e^{-2a} = \infty \), confirming that the function grows unboundedly as \( a \) gets very large negatively.