Problem 5 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. Let \(f(x)=\left(\frac{1}{3}\right)^{x} .\) As \(x \rightarrow \infty, f(x) \rightarrow\)__________.

Step-by-Step Solution

Answer

0

Step 1: Understand the problem

Here, a function is given as \(f(x)=\left(\frac{1}{3}\right)^{x}\). The goal is to find the value the function approaches as x approaches infinity.

Step 2: Observe the function

Notice that in the function \(f(x)=\left(\frac{1}{3}\right)^{x}\), as the value of x increases, the base to the power of x decreases, because the base is \(\frac{1}{3}\), a fraction less than 1.

Step 3: Apply the limit

In when x tends towards infinity, the function \(f(x)=\left(\frac{1}{3}\right)^{x}\) becomes a very small positive number, effectively it approaches zero. Thus the limit as \(x \rightarrow \infty\) is zero.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to the power of a variable exponent. They are commonly represented as f(x) = a^x, where a is the base and x is the exponent. When the base is greater than 1, the function generally grows rapidly as x increases. Conversely, if the base is a number between 0 and 1, as is the case with f(x) = (1/3)^x, the function decreases as x increases, drawing closer to 0.

One typical property of exponential functions is that they change at a rate proportional to their value, which is why they are often used to model growth or decay processes such as population growth or radioactive decay. A deeper understanding of exponential functions and their behavior aids significantly in subjects like biology, economics, and physics, where these natural processes can be represented and studied through mathematical models.

Limits of Functions

The concept of a limit is foundational in calculus and describes the behavior of a function as its argument approaches a specific value or, in some cases, infinity. To find the limit of a function like f(x) = (1/3)^x as x approaches infinity, we examine what value f(x) grows closer to as x gets larger and larger.

For example, in our exercise, as x becomes very large, the term (1/3)^x gets smaller because we are repeatedly multiplying a fraction less than 1 by itself, which will always yield a smaller number. Therefore, the limit of f(x) as x approaches infinity is 0. This concept is crucial in understanding how functions behave and in determining continuity, evaluating integrals, and solving differential equations.

Asymptotic Behavior

Asymptotic behavior examines how functions behave as they move towards infinity or some significant value, usually focusing on the end behavior of functions. In the context of the function f(x) = (1/3)^x, as x approaches infinity, f(x) approaches zero. The line y = 0 is then considered a horizontal asymptote of f(x).

A horizontal asymptote is a horizontal line that the graph of the function approaches but never actually reaches. It's important to note that approaching an asymptote doesn't always mean the function will reach the value zero exactly—just that it gets infinitely close. The concept of asymptotic behavior helps us predict the long-term trends of functions and can be essential when making long-term forecasts in fields such as finance and meteorology.