Question

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

Short answer

The limit \( \lim_{x \to c} f(x) = L \) is equivalent to \( \lim_{x \to c} [f(x)-L] = 0 \).

Step-by-step solution

Definition of Limit

Recall the definition of the limit: \( \lim_{x \to c} f(x) = L \) means that for every \( \epsilon > 0 \), there exists \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \), we have \( |f(x) - L| < \epsilon \).

Statement of the Problem

We need to prove that \( \lim_{x \to c} f(x) = L \) if and only if \( \lim_{x \to c} [f(x) - L] = 0 \). This involves proving two implications: one for each direction of the equivalence.

Prove \( \lim_{x \to c} f(x) = L \) implies \( \lim_{x \to c} [f(x) - L] = 0 \)

Suppose \( \lim_{x \to c} f(x) = L \). By definition, for every \( \epsilon > 0 \), there exists \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |f(x) - L| < \epsilon \). This directly means that \( |[f(x) - L] - 0| < \epsilon \), which shows that \( \lim_{x \to c} [f(x) - L] = 0 \).

Prove \( \lim_{x \to c} [f(x) - L] = 0 \) implies \( \lim_{x \to c} f(x) = L \)

Assume \( \lim_{x \to c} [f(x) - L] = 0 \). So, for every \( \epsilon > 0 \), there exists \( \delta > 0 \) such that if \( 0 < |x - c| < \delta \), then \( |[f(x) - L] - 0| < \epsilon \), or equivalently, \( |f(x) - L| < \epsilon \). Hence, by definition of the limit, \( \lim_{x \to c} f(x) = L \).

Conclusion

Since we proved both implications, \( \lim_{x \to c} f(x) = L \) if and only if \( \lim_{x \to c} [f(x) - L] = 0 \).

Key concepts

Epsilon-Delta Definition

The Epsilon-Delta definition is a fundamental aspect of understanding limits in calculus. It gives a precise way to define what it means for a function to approach a certain value, called a limit, as its input approaches some point. The definition is grounded in the use of two Greek letters and goes like this:- For any small, positive value \( \epsilon \) (no matter how tiny!), there exists another small, positive value \( \delta \).- Whenever the distance between \( x \) and \( c \) (written as \( |x - c| \)) is smaller than \( \delta \), but not zero, then the distance between \( f(x) \) and \( L \) (\( |f(x) - L| \)) must be smaller than \( \epsilon \). This means we're creating a tiny window around our input point \( c \) and the limit \( L \), wherein if one is satisfied, the other automatically follows.The Epsilon-Delta definition can initially seem abstract, but keep in mind, it's all about ensuring that as \( x \) sneaks near \( c \), \( f(x) \) huddles close to \( L \). This principle simplifies the process of proving equivalences, like the one given in the exercise.

Mathematical Proof

Mathematical proof is like a solid backbone that supports the truth of mathematical statements. It's a step-by-step process that carefully justifies why a claim is true or false. In the context of the given problem, the proof involves two directional implications:- First, we demonstrate that if the limit \( \lim_{x \to c} f(x) = L \) is true, then it leads to \( \lim_{x \to c} [f(x) - L] = 0 \).- Next, we need to prove the reverse, i.e., if \( \lim_{x \to c} [f(x) - L] = 0 \), it leads to \( \lim_{x \to c} f(x) = L \).Each direction of the proof relies on the Epsilon-Delta definition. For a mathematical proof, it's crucial not to skip steps, ensuring every logic leap is justified by concrete evidence based on a definition or property. By completing both directions, we establish the equivalence in a thorough, logical manner. The proof grants us the confidence to claim a mutual truth between two statements.

Equivalence of Statements

Understanding the equivalence of statements is crucial when we deal with proving limits and calculus concepts. Equivalence, in mathematical terms, implies that two statements are true in relation to each other—one cannot be true without the other being true as well. In logical terms, this means if statement A holds, statement B does too, and vice versa.With our problem, the equivalence means that knowing \( \lim_{x \to c} f(x) = L \) directly correlates with \( \lim_{x \to c} [f(x) - L] = 0 \). Neither can exist independently because they describe essentially the same scenario under different contexts. This equivalence is important as it simplifies analysis and ensures the consistency of our understanding of limits.In mathematics, demonstrating such equivalence involves showing both directions of implications meet assurance (if-and-only-if proof), which then makes both propositions interchangeable and robustly verified by their mutual dependency.