Problem 49 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(f(x) g(x)=1\) for all \(x\) and \(\lim _{x \rightarrow a} g(x)=0\). Prove that \(\lim _{x \rightarrow a} f(x)\) does not exist.

Step-by-Step Solution

Answer

The limit of \( f(x) \) as \( x \) approaches \( a \) does not exist because \( f(x) = \frac{1}{g(x)} \) approaches infinity.

Step 1: Understand the Given Equation

The equation given is \( f(x) g(x) = 1 \) for all \( x \). This relationship implies that as \( g(x) \) approaches 0, \( f(x) \) must approach infinity in some way to satisfy the equation, because if \( g(x) \) becomes very small, \( f(x) \) must become very large to compensate and keep the product as 1.

Step 2: Analyze the Limit of \( g(x) \)

It's given that \( \lim_{x \rightarrow a} g(x) = 0 \). This tells us that as \( x \) gets close to \( a \), the value of \( g(x) \) approaches zero.

Step 3: Derive the Implication for \( f(x) \)

Since \( f(x) g(x) = 1 \) for all \( x \), rearranging gives \( f(x) = \frac{1}{g(x)} \). To satisfy \( \lim_{x \rightarrow a} f(x) \), as \( g(x) \) approaches 0, \( \frac{1}{g(x)} \) approaches infinity. This implies \( f(x) \) becomes very large as \( x \) approaches \( a \).

Step 4: Conclude About the Existence of \( \lim_{x \rightarrow a} f(x) \)

For a limit to exist, \( f(x) \) must approach a particular finite value as \( x \) approaches \( a \). Since \( f(x) = \frac{1}{g(x)} \) approaches infinity, \( \lim_{x \rightarrow a} f(x) \) does not settle to a finite number. Therefore, the limit does not exist.

Key Concepts

Infinite Limits

When we talk about infinite limits, we are exploring scenarios where a function increases or decreases without bound as the input approaches a particular value. Let's take the function in our problem:

• We are given that as \(x\) approaches \(a\), \(g(x)\) approaches 0.
• This implies that \(f(x)\) must "compensate" to maintain the product \(f(x)g(x)=1\).
• Therefore, \(f(x)\) approaches an infinite value as \(g(x)\) approaches zero.

This is a classic example of an infinite limit: as one component in a product goes to zero, the other must go to infinity to ensure the product remains constant. Infinite limits help us understand and predict the behavior of functions that do not tend to a finite number.

Limit Laws

Limit laws are essential rules that provide a framework for finding the limits of functions, especially when combinations like sums, differences, products, or quotients are involved. They're like the grammar rules of calculus and guide us through complex limit problems painlessly. Here are some key aspects:

• Product Rule: The limit of a product is the product of the limits, if they exist. But note that this doesn't hold if one term tends to zero while the other goes to infinity.
• The law fails when dealing with limits of the type \(0\cdot\infty\), which we see in our exercise.
• In our case, \(f(x)\) approaches infinity as \(g(x)\) approaches zero, which disrupts the usual application of the product rule.

By understanding limit laws, we gain the ability to manipulate and simplify difficult limits, although, as shown here, it's crucial to be aware of the exceptions.

Indeterminate Forms

Indeterminate forms arise in calculus when an expression is not directly solvable due to an undefined form, such as \( \frac{0}{0} \) or \(0 \cdot \infty\). In this problem, as \(g(x)\) approaches zero, the expression \(f(x) = \frac{1}{g(x)}\) fits the indeterminate form \( \infty \cdot 0 \).
This form indicates that while we know \(f(x)\) will reach a very large number due to \(g(x)\) nearing zero, the exact behavior requires further analysis or numerical limits to resolve conclusively.

• An indeterminate form indicates more information is needed to understand the true behavior.
• Such a form often suggests oscillating, diverging, or growing behavior.

Understanding indeterminate forms allows us to correctly analyze these tricky expressions and clarify whether a limit exists or behaves infinitely, as seen here.

Product of Functions

The product of functions is crucial when evaluating limits, particularly when one function inversely relates to another, as seen with \(f(x)g(x)=1\) in our problem.

• Here, each function influences the behavior of the other, forcing \(f(x)\) to grow as \(g(x)\) shrinks to zero.
• The product being constant (1) adds another layer of complexity because it asks \(f(x)\) and \(g(x)\) to "balance out" their opposing trends.
• This relationship fundamentally restricts the typical application of product limit laws.

By examining how the functions interact through their product, one can infer necessary behavior – resulting either in finite or infinite outcomes. This interplay is a vital skill in calculus, helping students understand and predict complex function behaviors accurately. These concepts together showcase why \(\lim_{x\rightarrow a} f(x)\) does not exist, as \(f(x)\) cannot stabilize at a finite value.