Problem 13 - Calculus: Early Transcendentals

Question

Determine the following limits at infinity. $$\lim _{t \rightarrow \infty} e^{t}, \lim _{t \rightarrow-\infty} e^{t}, \text { and } \lim _{t \rightarrow \infty} e^{-t}$$

Step-by-Step Solution

Answer

Answer: As t approaches infinity, the limits are: - e^t: ∞ - e^(-t): 0 As t approaches negative infinity, the limit for e^t is 0.

Step 1: Determine the limit as t -> infinity for e^t

To find the limit for the exponential function as t approaches infinity, we can observe the behavior of the function. As t becomes larger, the value of e^t will also increase without bound because e > 1 (approximately 2.718). Therefore, the limit as t approaches infinity for e^t is: $$\lim_{t \rightarrow \infty} e^t = \infty$$

Step 2: Determine the limit as t -> -infinity for e^t

Now let's find the limit for the exponential function as t approaches negative infinity. As t becomes more negative, the value of e^t gets closer to 0. This is because as the exponent becomes a very large negative value, e^t essentially becomes 1 divided by a very large positive value, which makes it approach 0. Therefore, the limit as t approaches negative infinity for e^t is: $$\lim_{t \rightarrow-\infty} e^{t} = 0$$

Step 3: Determine the limit as t -> infinity for e^(-t)

Finally, let's find the limit for the exponential function with a negative exponent as t approaches infinity. As t becomes larger, the value of e^(-t) gets closer to 0. This is because the exponent is negated and essentially becomes more negative, making e^(-t) approach 0 just like in the previous case. Therefore, the limit as t approaches infinity for e^(-t) is: $$\lim_{t \rightarrow \infty} e^{-t} = 0$$ In conclusion, the limits for the given exponential functions are as follows: $$\lim_{t \rightarrow \infty} e^t = \infty$$ $$\lim_{t \rightarrow-\infty} e^{t} = 0$$ $$\lim_{t \rightarrow \infty} e^{-t} = 0$$

Key Concepts

Exponential Functions

Exponential functions are mathematical functions of the form \(f(t) = a^t\), where \(a\) is a constant and \(t\) is a variable. These functions have interesting properties because they grow very rapidly once the value of \(t\) increases. When \(a > 1\), the function experiences exponential growth, meaning the value of \(f(t)\) increases faster and faster as \(t\) becomes large.

• For \(e^t\), where \(e\) (approximately 2.718) is the base of the natural logarithm, it is a common example of an exponential function.
• Exponential functions are used to model many real-world situations, like population growth and compound interest.
• The rapid increase or decrease makes exponential functions crucial in both mathematical theory and applications.

Asymptotic Behavior

The term "asymptotic behavior" refers to the way a function behaves as the variable grows very large in magnitude. In other words, it describes how a function approaches a line, curve, or another function as the value of the variable heads towards infinity or negative infinity.

• For \(e^t\), as \(t\) heads to infinity, \(e^t\) becomes infinitely large, demonstrating rapid growth.
• When \(t\) tends towards negative infinity, \(e^t\) approaches zero, as it becomes an inverse of a large positive number.
• Similarly, \(e^{-t}\) approaches zero as \(t\) moves towards infinity, because negating the exponent flips the growth direction of the function.

Understanding asymptotic behavior is essential for comprehending how different functions behave at extreme values, which is useful in fields like physics and engineering.

Limit Concept

The limit concept in calculus is a fundamental idea that helps us understand and analyze the behavior of functions as the input approaches a particular value. It is often used to describe the value that a function "approaches" as the input changes.

• In our exercise, we deal with limits as \(t\) approaches both positive and negative infinity.
• The limit \(\lim_{t \rightarrow \infty} e^t = \infty\) means that the function grows without bound as \(t\) becomes very large.
• Conversely, the limit \(\lim_{t \rightarrow -\infty} e^t = 0\) indicates that the function approaches zero as \(t\) becomes very negative.
• Lastly, \(\lim_{t \rightarrow \infty} e^{-t} = 0\) suggests the function approaches zero for very large values of \(t\), due to flipping the rapid growth to rapid decay.

Limits are not just abstract concepts; they represent real tendencies in nature and technology, where they help predict long-term trends and outcomes.