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Magazine Chapter 2: Problem 37
Prove that \(\lim _{x \rightarrow c} f(x)=L \Leftrightarrow \lim _{x \rightarrow c}[f(x)-L]=0\).
Short Answer
Expert verified
By definition of limits, both expressions imply each other, proving their equivalence.
Step by step solution
01
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 \(|f(x) - L| < \epsilon\).
02
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.
03
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 \( \lim_{x \to c} [f(x) - L] = 0 \).
04
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: \( \lim_{x \to c} f(x) = L \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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鈥檚 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 \( \epsilon \).
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.
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 \( \lim_{x \to c} f(x) = L \).
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.
