Problem 66 - A Graphical Approach to Precalculus with Limits

Question

Find two functions \(f(x)\) and \(g(x)\) with the given properties. $$\lim _{x \rightarrow \infty} f(x)=0, \lim _{x \rightarrow \infty} g(x)=\infty, \text { and } \lim _{x \rightarrow \infty}[f(x) \cdot g(x)]=0$$

Step-by-Step Solution

Answer

\( f(x) = \frac{1}{x^2} \), \( g(x) = x \) satisfy all conditions.

Step 1: Understanding the Problem

We need to find two functions \( f(x) \) and \( g(x) \) such that \( \lim _{x \rightarrow \infty} f(x)=0 \), \( \lim _{x \rightarrow \infty} g(x)=\infty \), and \( \lim _{x \rightarrow \infty}[f(x) \cdot g(x)]=0 \). Essentially, as \( x \) approaches infinity, \( f(x) \) should approach zero, \( g(x) \) should increase without bound, and their product should still approach zero.

Step 2: Choosing a Form for \( f(x) \)

A common choice for a function that approaches 0 as \( x \rightarrow \infty \) is a function that decays, such as \( f(x) = \frac{1}{x} \). This function gets smaller as \( x \) becomes larger.

Step 3: Choosing a Form for \( g(x) \)

We need a function that increases without bound as \( x \rightarrow \infty \). One simple choice is \( g(x) = x \) because it grows linearly and approaches infinity.

Step 4: Calculate \( f(x) \cdot g(x) \)

With our chosen functions, we calculate their product: \( f(x) \cdot g(x) = \frac{1}{x} \cdot x = 1 \).

Step 5: Verify Limits of Product

To satisfy all conditions, \( \lim _{x \rightarrow \infty}[f(x) \cdot g(x)] = 1 \). This is incorrect based on our needs, so the function forms are inappropriate. And instead considering \( f(x) = \frac{1}{x^2} \) instead to adjust the decay rate such that \( f(x) \cdot g(x) \rightarrow 0 \).

Step 6: Re-evaluate Product with New Form

Let's try \( f(x) = \frac{1}{x^2} \) and \( g(x) = x \). Then: \( f(x) \cdot g(x) = \frac{1}{x^2} \cdot x = \frac{1}{x} \).

Step 7: Check Limit of New Product

Calculate the limit of the new product: \( \lim _{x \rightarrow \infty} \frac{1}{x} = 0 \). This satisfies the third condition as the product approaches 0.

Key Concepts

Functions Approaching Limits

In calculus, the concept of limits is vital for understanding the behavior of functions as they approach certain values, particularly infinity. When we say a function like \( f(x) \) approaches a limit, it means as \( x \) becomes infinitely large, \( f(x) \) approaches a specific value. This isn't always an exact match to a number, but rather a trend.
For example, the function \( f(x) = \frac{1}{x} \) approaches 0 as \( x \rightarrow \infty \). This is because as \(x\) increases, the fraction \( \frac{1}{x} \) becomes smaller and smaller. Here are some other key points about functions approaching limits:

• Limits help us analyze behaviors at boundaries or approaching large values.
• They're foundational in defining derivatives and integrals.
• Understanding limits builds intuition around function behavior as variables change extensively.

Recognizing how functions approach limits allows us to predict outcomes and construct new functions that behave in desired ways.

Infinite Limits

Infinite limits explore what happens as a function grows without bound; essentially when it becomes infinitely large. This can occur either positively or negatively. For example, consider \( g(x) = x \). As \( x \rightarrow \infty \), \( g(x) \) also heads towards infinity because it continues to grow larger with increasing \( x \). Infinite limits are important because they help us identify where a function will not converge to a specific value but instead increase or decrease indefinitely. Here are a few essential points about infinite limits:

• They describe situations where functions don't settle at a finite value.
• These limits help identify vertical asymptotes and end-behavior of functions.
• They are primarily concerned with the growth rate and direction rather than specific numerical limits.

Understanding infinite limits can apply many real-world phenomena where calculations tend to unbound conditions, like exponential growth scenarios.

Product of Functions

The product of functions pertains to multiplying two or more functions to see the resulting behavior when combined. For instance, in the exercise, we consider \( f(x) = \frac{1}{x^2} \) and \( g(x) = x \). Their product is \( f(x) \cdot g(x) = \frac{1}{x} \), showcasing how the combination affects the overall limit as \( x \rightarrow \infty \). When dealing with products of functions, consider how each function behaves individually and the cumulative effect:

• The behavior of individual functions impacts the combined product.
• Products can reveal stabilization, cancellation, or amplify certain aspects of functioning limits.
• Understanding how to manipulate function forms to achieve desired outcomes in their limits is valuable.

Mastery of the product of functions allows one to solve complex real-life problems, ensuring a comprehensive perspective on growth behaviors and outcomes as they relate to varying inputs.