Problem 4 - 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

As \(x\) approaches \(-\infty\), \(f(x)\) approaches 0.

Step 1: Identify the Function

The function given is \(f(x) = 5e^{x}\), which is an exponential function in base \(e\).

Step 2: Recognize the Behavior of Exponential Functions

The behavior of the exponential function \(e^{x}\) is such that as \(x\) approaches \(-\infty\), \(e^{x}\) approaches 0.

Step 3: Apply Function Transformation

The function \(f(x) = 5e^{x}\) is a vertical scaling of the function \(e^{x}\) by a factor of 5. However, this scaling does not affect the limit as \(x\) approaches \(-\infty\). Multiplying by 5 does not change the fact that \(e^{x}\) approaches 0 as \(x\) approaches \(-\infty\).

Key Concepts

Limit of Exponential Function

Understanding the limit of an exponential function is crucial when analyzing its behavior at extreme values. Consider the function f(x) = 5e^x. As x tends towards negative infinity (x → -∞), the value of e^x gets closer and closer to 0. This happens because exponential functions grow very quickly in the positive direction, while they approach zero (but never reach it) as they go towards negative infinity. So, for the given function f(x), as x → -∞, f(x) also approaches 0, regardless of the initial scalar multiplication by 5. The correct completion of the exercise would be: As x → -∞, f(x) → 0.

The intuition behind this is simple: any constant multiplied by a value getting infinitely small will result in a number incredibly close to zero—hence, the limit of f(x) as x approaches negative infinity is 0.

Exponential Function Transformation

Exponential function transformation involves operations that alter the appearance or position of the graph of the base function without changing its fundamental behavior. Taking the base function e^x, scaling it vertically by a constant doesn't affect the property that as x → -∞, the function's value approaches zero. Instead, the graph of the function f(x) = 5e^x would be stretched vertically, expanding away from the x-axis at a faster rate than e^x alone. However, the exponential decay towards zero as x → -∞ remains intact.

Transformations such as these—including scaling, shifting, and reflecting—are critical for students to visualize how functions behave and to understand how different parameters affect the graph of the function.

Asymptotic Behavior of Functions

Asymptotic behavior of functions refers to how a function acts as the input gets larger (either positively or negatively). The term 'asymptote' itself describes a line that the graph of a function approaches but never actually touches. For an exponential function like f(x) = 5e^x, the x-axis (or y = 0) acts as a horizontal asymptote when x is negatively large.

It's important to note that while the values of f(x) get very, very close to zero, they are never exactly zero; this unique trait is a staple characteristic of asymptotic behavior. When dealing with the concept of limits, this asymptotic nature is often the key to finding the right answer, as it steers the function towards a predictable path as the inputs grow in magnitude.