Problem 65 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

Choose \(f(x) = \frac{1}{x}\) and \(g(x) = x^2\).

Step 1: Analyze the Limits

We need two functions, \(f(x)\) and \(g(x)\), such that \(\lim_{x \to \infty} f(x) = 0\), \(\lim_{x \to \infty} g(x) = \infty\), and \(\lim_{x \to \infty} [f(x) \cdot g(x)] = \infty\). Essentially, \(f(x)\) must approach zero, while \(g(x)\) grows infinitely, yet the product \(f(x) \cdot g(x)\) also grows infinitely.

Step 2: Choose a Suitable Function for f(x)

A function that approaches zero as \(x\) approaches infinity is \(f(x) = \frac{1}{x}\). This function satisfies \(\lim_{x \to \infty} \frac{1}{x} = 0\).

Step 3: Choose a Suitable Function for g(x)

A function that grows without bound as \(x\) approaches infinity is \(g(x) = x^2\). This function satisfies \(\lim_{x \to \infty} x^2 = \infty\).

Step 4: Verify the Product Limit Condition

Now, let's verify the condition for the product. Calculate \(\lim_{x \to \infty} \left( \frac{1}{x} \cdot x^2 \right) = \lim_{x \to \infty} x = \infty\). The product grows infinitely, fulfilling the condition.

Step 5: Conclusion

By choosing \(f(x) = \frac{1}{x}\) and \(g(x) = x^2\), we satisfy all three requirements: \(f(x)\) approaches zero, \(g(x)\) grows infinitely, and \(f(x) \cdot g(x)\) grows infinitely.

Key Concepts

Function Behavior

In calculus, the behavior of a function describes how it changes as the input variable moves towards a particular value, often infinity. To better understand this, suppose we have two functions: \(f(x) = \frac{1}{x}\) and \(g(x) = x^2\).

• For \(f(x) = \frac{1}{x}\), as \(x\) becomes larger, the value of \(f(x)\) becomes smaller and smaller, approaching zero. This indicates that \(f(x)\) has a decreasing trend as \(x\) approaches infinity.
• On the other hand, \(g(x) = x^2\) acts quite differently. As \(x\) increases, \(g(x)\) also increases rapidly, suggesting that \(g(x)\) is an increasing function as \(x\) heads towards infinity.

By analyzing individual behaviors of these functions, one gains insight into their long-term tendencies and overall behavior. Understanding such behaviors is crucial when considering the interaction and product of several functions.

Infinity in Calculus

Infinity plays a significant role in calculus, often used to describe unbounded behaviors or tendencies. When we say \(\lim_{x \to \infty} f(x) = 0\), it refers to the function approaching a particular value as \(x\) becomes indefinitely large.

• In our exercise, \(f(x) = \frac{1}{x}\) approaches 0 because the fraction gets smaller as the denominator \(x\) grows.
• Conversely, \(\lim_{x \to \infty} g(x) = \infty\) signifies that \(g(x) = x^2\) increases without bound since larger values of \(x\) result in even larger outputs.

Infinity helps capture the idea of limitless growth or shrinkage, which is vital when examining function behaviors over large ranges. The notation handles such conditions elegantly in calculus, providing a way to analyze seemingly unmanageable scenarios.

Product of Functions

The concept of the product of functions involves combining two or more functions by multiplying them. This interaction can lead to unique behaviors, some of which were highlighted in our problem where \(f(x) = \frac{1}{x}\) and \(g(x) = x^2\) were chosen. Let's discuss the result of their multiplication:

• When multiplied, \(f(x) \cdot g(x) = \frac{1}{x} \cdot x^2 = x\). This combination transforms our initial functions into a product which grows infinitely as \(x\) increases.
• Even though \(f(x)\) tends to zero, \(g(x)\)'s rapid increase dominates, ensuring the product grows without bound, i.e., \(\lim_{x \to \infty} f(x) \cdot g(x) = \infty\).

This demonstrates how multiplying functions can yield results that highlight the dominant behavior of one function over another. It is essential to explore these interactions to see how functions can complement or contradict each other in calculus.