Problem 3 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. Let \(f(x)=5 e^{x} .\) As \(x \rightarrow \infty, f(x) \rightarrow\)__________.

Step-by-Step Solution

Answer

\(\lim_{{x \to \infty}} f(x) = \infty\).

Step 1: Explore the function

First, let's look closer to the given function \(f(x)=5e^x\). This is an exponential function where \(e\) is Euler's number, which is approximately equal to 2.71828.

Step 2: Understand the behavior of exponential function.

The nature of exponential functions is such that the function will increase as the value of \(x\) increases. For \(f(x)=e^x\), as \(x\) approaches infinity, \(e^x\) also approaches infinity. Since the given function is \(f(x)=5e^x\), without any negative sign or term subtracting from it, as \(x\) approaches infinity, \(f(x)\) will also approach infinity.

Step 3: Calculate the limit.

The limit of \(f(x)=5e^x\) as \(x\) approaches infinity is therefore infinity. We can write it formally as \(\lim_{{x\to \infty}} 5e^x = \infty\).

Key Concepts

Limits

In mathematics, a limit describes the value that a function approaches as the input approaches a certain point. Limits are essential for understanding the behavior of functions, especially as the inputs grow larger or smaller. In our exercise, we use limits to determine what happens to the function \(f(x) = 5e^x\) as \(x\) gets very large, specifically as \(x\) approaches infinity. The notation

• \(\lim_{{x\to \infty}} 5e^x\)

helps us express this idea concisely. When \(x\) becomes very large, the function \(5e^x\) keeps growing because \(e^x\) becomes larger and larger. Therefore, the limit of \(5e^x\) as \(x\) approaches infinity is infinity itself. This illustrates how limits offer a way to grasp what happens at the extremes of a function's domain, even when we can't compute an exact value.

Asymptotic Behavior

Asymptotic behavior gives us a deeper understanding of how functions act when inputs become extremely large or small. For exponential functions like \(f(x) = 5e^x\), their asymptotic behavior as \(x\) approaches infinity is critical. These functions grow very quickly, which we refer to as having an asymptote at infinity.An asymptote is a line that functions approach but never actually reach. In the case of \(5e^x\), as \(x\) grows, its asymptotic behavior shows a sharp increase toward infinity. Unlike other types of functions, like rational or polynomial functions that might level off or approach a horizontal line as \(x\) becomes large, exponential functions do not have horizontal asymptotes.Thus, we see that the asymptotic behavior of \(5e^x\) is steep and unbounded as \(x\) approaches infinity, highlighting the incredibly rapid growth rate of exponential functions.

Function Behavior

Function behavior refers to how a function acts or changes as \(x\) varies across its domain. For \(f(x) = 5e^x\), understanding its behavior means analyzing how it increases as \(x\) increases.Exponential functions are known for their rapid growth. As \(x\) grows, \(e^x\) increases dramatically, and being multiplied by 5 makes the growth of \(f(x)\) five times greater. There's no slowing down—the function's value keeps rising steadily without bounds.This critical behavior is evident from the graph of the function as well: it's a curve that rises sharply upwards as \(x\) extends to infinity. The function does not have peaks, valleys, or turning points; it just continuously increases. Moreover, such behavior is crucial in real-world applications, like modeling population growth or radioactive decay, where understanding how quickly a quantity is increasing or decreasing over time is important.