Problem 101 - Single Variable Calculus

Question

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that shows it is false. If \(\lim _{x \rightarrow a a}[f(x)-g(x)]\) exists, then \(\lim _{x \rightarrow a} f(x)\) and \(\lim _{x \rightarrow a} g(x)\) also exist.

Step-by-Step Solution

Answer

The given statement is **false**. A counterexample is \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{1}{x}\). While the limit of the difference function \(f(x) - g(x)\) exists as \(x \rightarrow 0\), the individual limits of \(f(x)\) and \(g(x)\) do not exist as \(x\) approaches 0.

Step 1: Analyze the claim

First, let's analyze the given statement. We are given that the limit of the difference function \(f(x) - g(x)\) exists as \(x\) approaches \(a\). Our task is to determine if this necessarily implies that the limit of each individual function, i.e., \(f(x)\) and \(g(x)\), exists as \(x \to a\). Remember that, if the limit of the difference function \(f(x) - g(x)\) exists as \(x\) approaches \(a\), then we can write: \( \lim_{x \rightarrow a} [f(x) - g(x)] = L \) for some finite value \(L\). Now, we need to examine if this is sufficient to conclude that the individual limits of \(f(x)\) and \(g(x)\) exist.

Step 2: Check properties of limits

Recall the properties of limits. One of them states that if both the limit of \(f(x)\) and the limit of \(g(x)\) as x goes to a exist, then \( \lim_{x \rightarrow a} [f(x) - g(x)] = \lim_{x \rightarrow a} f(x) - \lim_{x \rightarrow a} g(x) \) However, this property does not guarantee the converse, which the given statement claims that if \( \lim_{x \rightarrow a} [f(x) - g(x)] \) exists, then \( \lim_{x \rightarrow a} f(x) \) and \( \lim_{x \rightarrow a} g(x) \) also exist.

Step 3: Provide a counterexample

In order to show that the given statement is false, we can provide a counterexample. Consider the two functions, \(f(x) = \frac{1}{x}\) and \(g(x) = \frac{1}{x}\). We know that neither limit exists as \(x\) approaches 0: \( \lim_{x \rightarrow 0} f(x) = \lim_{x \rightarrow 0} \frac{1}{x} \) does not exist \( \lim_{x \rightarrow 0} g(x) = \lim_{x \rightarrow 0} \frac{1}{x} \) does not exist However, when computing the difference function \(f(x) - g(x)\), we have: \(f(x) - g(x) = \frac{1}{x} - \frac{1}{x} = 0\) So, the limit of the difference function does exist as \(x\) approaches 0: \( \lim_{x \rightarrow 0} [f(x) - g(x)] = \lim_{x \rightarrow 0} 0 = 0 \) This counterexample shows that even if the limit of the difference function exists, the individual limits of the functions may not exist. Therefore, the statement is **false**.

Key Concepts

Existence of Limits

When we talk about the existence of limits in calculus, we're trying to determine if a function approaches a specific value as the input approaches some point. For a function like \( f(x) \), we say the limit exists at \( x = a \) if \( f(x) \) gets arbitrarily close to some number \( L \) as \( x \) gets closer to \( a \). This is represented mathematically as \( \lim_{x \to a} f(x) = L \).
Here's a simple way to think about it:

• If you can predict the output of the function \( f(x) \) as \( x \) approaches \( a \), the limit exists.
• If \( f(x) \) behaves unpredictably or doesn't approach a single value, the limit does not exist.

Understanding the existence of limits helps us visualize and predict the behavior of functions in calculus. In our context, just because the limit of a difference \( f(x) - g(x) \) exists doesn't necessarily mean the individual limits \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) exist as well.

Limit Properties

Limit properties are rules that help simplify the calculation of limits. One common property is the difference rule, which states that if both \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) exist, then \( \lim_{x \to a} [f(x) - g(x)] \) equals \( \lim_{x \to a} f(x) - \lim_{x \to a} g(x) \).
Some helpful properties of limits:

• Sum/Difference Rule: The limit of a sum or difference is the sum or difference of the limits.
• Product Rule: The limit of a product is the product of the limits.
• Quotient Rule: The limit of a quotient is the quotient of the limits (provided the denominator is not zero).

These properties are nonetheless not reversible. Having the limit of \( f(x) - g(x) \) exist doesn't imply that both individual limits exist. Our exercise demonstrates this through the function \( \frac{1}{x} \) where each separate limit does not exist, yet their difference does.

Counterexample in Calculus

Counterexamples are powerful tools for disproving statements in calculus. They provide specific examples where a general claim fails to hold true. In our exercise, we show that the claim "if \( \lim_{x \to a} [f(x) - g(x)] \) exists, then both \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) exist" is false by using a counterexample.
Consider the functions \( f(x) = \frac{1}{x} \) and \( g(x) = \frac{1}{x} \):

• Neither \( \lim_{x \to 0} f(x) \) nor \( \lim_{x \to 0} g(x) \) exist because \( \frac{1}{x} \) becomes infinite as \( x \) approaches zero.
• Yet, their difference \( \frac{1}{x} - \frac{1}{x} = 0 \) has an existing limit of zero.

This counterexample helps illustrate that even if a limit of a difference exists, the individual functions' limits may not. Such examples are essential for grasping the subtleties and intricacies of calculus limits.