Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove that \(\lim _{x \rightarrow c} f(x)=L \Leftrightarrow \lim _{x \rightarrow c}[f(x)-L]=0\).

Step-by-Step Solution

Answer

By definition of limits, both expressions imply each other, proving their equivalence.

Step 1: Understand the Definition of Limit

The limit of a function \(f(x)\) as \(x\) approaches \(c\) is \(L\) if for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), it follows that \(|f(x) - L| < \epsilon\).

Step 2: Reformulate the Statement

We want to prove that \( \lim_{x \to c} f(x) = L \) is equivalent to \( \lim_{x \to c} [f(x) - L] = 0 \). This means showing that both expressions imply one another by the definition of a limit.

Step 3: Prove Limit Implication from Original

Assume \( \lim_{x \to c} f(x) = L \). This implies that \( |f(x) - L| < \epsilon \) for \(0 < |x - c| < \delta\), where \(\delta\) is dependent on \(\epsilon\). If you define a new function \(g(x) = f(x) - L\), then \( |g(x) - 0| = |f(x) - L| < \epsilon\). This shows \( \lim_{x \to c} [f(x) - L] = 0 \).

Step 4: Prove Reverse Implication

Assume \( \lim_{x \to c} [f(x) - L] = 0 \). This means for every \(\epsilon > 0\), there exists a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), \(|f(x) - L| < \epsilon\). Therefore, with \(f(x) - L = g(x)\), the original statement is achieved: \( \lim_{x \to c} f(x) = L \).

Step 5: Conclude the Equivalence

Both implications are true: \( \lim_{x \to c} f(x) = L \) implies \( \lim_{x \to c} [f(x) - L] = 0 \) and vice versa. Thus, we have shown the equivalence by using the definition of the limit in both directions.

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is a formal approach to understanding limits. It is a way to say that as we get very close to a point, the values of a function come very close to a specific number. Here’s how it works:

• For every small positive number called \( \epsilon \), we can find another small positive number \( \delta \).
• This \( \delta \) has the property that whenever the distance between \( x \) and \( c \) (the point we are approaching) is less than \( \delta \), the distance between \( f(x) \) and \( L \) (the limit) is less than \( \epsilon \).

In simpler terms, this definition helps ensure that as \( x \) gets closer to \( c \), \( f(x) \) gets closer to \( L \). This forms the cornerstone for proving equivalence between limits of functions and their variations.

Function

A function in mathematics is essentially a rule that takes an input and gives an output. For each input, the function assigns one and only one output.

• Example: The function \( f(x) = x^2 \) takes a number \( x \) and outputs its square.

When working with limits, functions help us understand how values change as we approach particular points. In our exercise, we are looking at \( f(x) \) and examining its behavior as we get closer to a value \( c \). Understanding the nature of functions helps in proving properties about limits.

Equivalence

Equivalence, in the context of limits, means two statements express the same truth. When we say \( \lim_{x \to c} f(x) = L \) is equivalent to \( \lim_{x \to c} [f(x) - L] = 0 \), we mean:

• If one statement is true, the other must also be true.
• This involves showing if \( \lim_{x \to c} f(x) = L \) holds, then automatically \( \lim_{x \to c} [f(x) - L] = 0 \) must hold.
• The reverse is also necessary: if \( \lim_{x \to c} [f(x) - L] = 0 \) is true, so is \( \lim_{x \to c} f(x) = L \).

Proving equivalence requires showing both directions: forward and reverse implications, which is vital in mathematical problems to ensure completeness.

Mathematical Proof

A mathematical proof is a logical argument that verifies the truth of a statement. Proofs often involve breaking down a statement into smaller, verifiable parts and using already known facts or axioms.

• In our context, we wanted to prove an equivalence regarding limits, which involves checking two implications.
• First, assume one side of the equivalence and demonstrate that the other side follows using the epsilon-delta definition.
• Then, reverse it: assume the second part of the equivalence, and prove the first statement is a consequence of it.

A good proof in mathematical terms leaves no doubt and confirms the relationship or property holds under specified conditions. Through this exercise, using proofs gives us confidence and verifies our understanding of limits in calculus.