Problem 6 - Applied Calculus

Question

1-16. Evaluate each limit (or state that it does not exist using \(\infty\) and \(-\infty\) where appropriate). $$ \lim _{x \rightarrow \infty}\left(1-e^{-x / 3}\right) $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Identify the parts of the function

The function given is \( f(x) = 1 - e^{-x/3} \). We need to evaluate \( \lim _{x \rightarrow \infty} (1 - e^{-x/3}) \). The important part of this function is the exponential term \( e^{-x/3} \).

Step 2: Evaluate the exponential term as x approaches infinity

As \( x \) approaches infinity, \( -x/3 \) approaches \( -\infty \). Therefore, \( e^{-x/3} \) approaches \( e^{- ext{large positive number}} \), which is \( 0 \).

Step 3: Substitute the limit of the exponential term

Since \( e^{-x/3} \to 0 \) as \( x \to \infty \), substitute this result into \( 1 - e^{-x/3} \). This gives us: \( 1 - 0 = 1 \).

Step 4: Conclusion of the limit

The limit of \( f(x) = 1 - e^{-x/3} \) as \( x \to \infty \) is \( 1 \). This concludes that as \( x \) becomes larger, the function approaches the value 1.

Key Concepts

Exponential Functions

Exponential functions are a cornerstone of calculus, representing rapid growth or decay models in various scientific fields. At the heart of these functions is the base, typically the mathematical constant \( e \), approximately equal to 2.718. This base \( e \) is central to natural exponential functions, denoted as \( e^x \), and it's unique because it is the only rate that is its own derivative.

In expressions like \( e^{-x/3} \), the exponent affects the growth or decay behavior of the function. Here, the negative sign indicates a decay, and dividing \( x \) by 3 slows down this decay rate. The exponential term \( e^{-x/3} \) shrinks towards zero as \( x \) grows larger. This behavior is vital when evaluating limits, as exponential terms can significantly influence the outcome of the function's behavior at infinity or zero.

Limit Evaluation

Limit evaluation helps us understand the behavior of functions as inputs approach certain values. In calculus, the limit of a function \( f(x) \) as \( x \) approaches a particular value, tells us where the function is "heading".

For example, evaluating \( \lim_{x \to \infty} (1 - e^{-x/3}) \) involves examining the behavior of the exponential component. As \( x \) becomes very large, the term \( e^{-x/3} \) approaches zero, because the negative exponent leads the expression towards decay. By substituting \( e^{-x/3} = 0 \) into the expression \( 1 - e^{-x/3} \), we determine the limit is 1. This means that as the input grows boundlessly large, the function's output approaches 1.

Infinity in Calculus

Infinity in calculus plays a crucial role in understanding limits and the behavior of functions at extreme values. When a limit involves \( x \to \infty \), it often evaluates what happens as \( x \) increases without bound.

Infinity can symbolize growth beyond any finite bounds or decay towards zero, depending on the function's structure. In our example, the limit \( \lim_{x \to \infty} (1 - e^{-x/3}) \) illustrates decay towards a finite value—in this case, the number 1—as \( x \) grows. Understanding this concept clarifies many real-world phenomena, such as chemical reactions approaching completion or cooling processes reaching ambient temperature. Infinity allows calculus to quantify and describe these abstract, boundless concepts effectively.