Problem 13 - APEX Calculus

Question

Prove the given limit using an \(\varepsilon-\delta\) proof. $$ \lim _{x \rightarrow 1} \frac{1}{x}=1 $$

Step-by-Step Solution

Answer

Since \(|x - 1| < \delta\) leads to \(\left| \frac{1}{x} - 1 \right| < \varepsilon\), the limit is proven.

Step 1: Understanding the Limit Statement

We need to prove using an \(\varepsilon-\delta\) proof that \(\lim _{x \rightarrow 1} \frac{1}{x} = 1\). This means that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that if \(|x - 1| < \delta\) then \(\left| \frac{1}{x} - 1 \right| < \varepsilon\).

Step 2: Express the Absolute Difference

Consider the absolute difference \(\left| \frac{1}{x} - 1 \right| = \left| \frac{1-x}{x} \right| = \frac{|1-x|}{|x|}\). To ensure this is less than \(\varepsilon\), choose \(\delta\) appropriately.

Step 3: Propose a Bound for \(x\)

Assume \(|x - 1| < \delta\), which implies \(1-\delta < x < 1+\delta\). To have \(x\) not getting too close to zero (since that would make \(\frac{1}{x}\) undefined or large), let's propose that \(\delta < 0.5\). This gives \(0.5 < x < 1.5\).

Step 4: Bound \(|x|\) Away from Zero

From the inequality \(0.5 < x < 1.5\), we have \(|x| \geq 0.5\) since the smallest positive \(x\) can be is 0.5. This bound helps control the troubling behavior near \(x = 0\).

Step 5: Inequality Manipulation

We now have \(\frac{|1 - x|}{|x|} < \varepsilon\). Given \(|x| \geq 0.5\), it follows that:\[|1 - x| < \varepsilon |x|\leq \varepsilon \cdot 1.5.\]Chose \(\delta\) so that \(\delta \leq \varepsilon \cdot 0.5\).

Step 6: Verify the Conditions with \(\delta\)

If \(\delta = \min\{0.5, \varepsilon \cdot 0.5\}\) is chosen, then for \(|x - 1| < \delta\), it ensures \(|x| \geq 0.5\) and:\[\frac{|1-x|}{|x|} < \varepsilon.\]This satisfies the \(\varepsilon-\delta\) condition.

Key Concepts

Understanding Limits in Calculus

Limits are foundational to calculus. They describe how a function behaves as the input approaches a certain value. The notation \( \lim_{x \to a} f(x) = L \) implies that as \( x \) approaches \( a \), the function \( f(x) \) gets arbitrarily close to \( L \). This is crucial because it allows us to understand function behaviors even at values where they might not be explicitly defined.
In our exercise, the limit we are considering is \( \lim_{x \to 1} \frac{1}{x} = 1 \). This signifies that as \( x \) gets closer to 1 from both sides, the value of \( \frac{1}{x} \) approaches 1.

• The role of limits involves exploring the "approaching behavior" rather than actual substitution.
• It helps in finding derivatives and evaluating integrals, which are the core operations in calculus.

Using the \( \varepsilon-\delta \) proof method to validate this limit means setting conditions for precision (\( \varepsilon \)) and closeness (\( \delta \)) to show this approach rigorously.

Grasping Continuity Through Limits

Continuity means that a function doesn’t have any jumps, breaks, or infinite holes. A function is continuous at a point \( a \) if \( \lim_{x \to a} f(x) = f(a) \).
This connects seamlessly with the concept of limits, as they help us define continuity at every point within a function.
In our example, showing \( \lim_{x \to 1} \frac{1}{x} = 1 \) using an \( \varepsilon-\delta \) proof also speaks about continuity around \( x=1 \). What it does is:

• Verifies the mathematical definition of a limit, implying the function is continuous around that point.
• Ensures the function value and the predicted limit match as \( x \) approaches 1.

Understanding continuity involves seeing that if you pick any value near \( a \), you won’t find a sudden change in the function's value. In essence, it’s about the function behaving in a predictable manner.

Real Analysis and Rigorous Proofs

Real Analysis might seem daunting, but it's simply the in-depth study of real-valued sequences and their limits. It lays out the rules and justifications for why calculus works.
In the case of our \( \varepsilon-\delta \) proof for \( \lim_{x \to 1} \frac{1}{x} = 1 \), we're applying real analysis to establish certainty. Real analysis doesn’t take shortcuts; every statement must be proven.

• It develops the idea of limits and continuity into a solid framework.
• Breaks down complex calculus concepts into logical, provable steps.
• Ensures mathematical results are foolproof by demanding proofs from basic axioms or previously established results.

The purpose is to substantiate claims within calculus by providing exhaustive proofs that confirm even the simplest calculus outcomes. Striving for rigor makes real analysis a crucial component of mathematics.