Question

Give an \(\varepsilon-\delta\) proof of each limit fact. $$ \lim _{x \rightarrow 1} \sqrt{2 x}=\sqrt{2} $$

Short answer

The limit is verified by setting \(\delta = \min\left(0.5, \frac{\varepsilon \sqrt{2}}{2}\right)\).

Step-by-step solution

Understand the Limit Statement

We need to prove that \(\lim_{x \rightarrow 1} \sqrt{2x} = \sqrt{2}\). This means that for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever \(|x - 1| < \delta\), it follows that \(|\sqrt{2x} - \sqrt{2}| < \varepsilon\).

Express the Problem in \(\varepsilon\) Terms

To satisfy \(|\sqrt{2x} - \sqrt{2}| < \varepsilon\), we'll simplify this inequality. We start by using the factored form of a difference of squares to express it as \(|\sqrt{2x} - \sqrt{2}| = \frac{2|x - 1|}{|\sqrt{2x} + \sqrt{2}|}\).

Simplify and Solve for \(\delta\)

To make \(\frac{2|x - 1|}{|\sqrt{2x} + \sqrt{2}|} < \varepsilon\), we can multiply both sides by \(|\sqrt{2x} + \sqrt{2}|\) (assuming it's not zero), obtaining \(2|x - 1| < \varepsilon \cdot |\sqrt{2x} + \sqrt{2}|\).

Estimate \(|\sqrt{2x} + \sqrt{2}|\)

Since \(x\) is near 1, \(\sqrt{2x}\) is near \(\sqrt{2}\). We can bound \(|\sqrt{2x} + \sqrt{2}|\) below by considering it to be close to \(2\sqrt{2}\), which holds for \(|x-1|<0.5\). Therefore, \(|\sqrt{2x} + \sqrt{2}| > \sqrt{2}\).

Choose Values for \(\delta\)

Now choose \(\delta = \min\left(0.5, \frac{\varepsilon \sqrt{2}}{2}\right)\). This ensures \(2|x - 1| < \varepsilon \cdot \sqrt{2}\) whenever \(|x - 1| < \delta\).

Verify the Proof

For \(|x - 1|<\delta\), the expression \(2|x - 1| < \varepsilon \sqrt{2}\) holds, showing that \(|\sqrt{2x} - \sqrt{2}| < \varepsilon\). This satisfies the limit definition and completes the \(\varepsilon-\delta\) proof.

Key concepts

Limit Theorem

The limit theorem is a fundamental concept in calculus. It tells us how a function behaves as we approach a certain point from either direction. In the exercise, the limit is investigated as the variable approaches 1.
Note that as the variable gets close to the limit point, the function's value gets closer to the limit value specified—in this case, \(\sqrt{2}\).
Understanding limits helps us comprehend the behavior of functions at boundary points or points of interest where direct evaluation is challenging or impossible.

• The limit can be thought of as a formalized way of speaking about what happens near a point.
• They help define derivatives and integrals, offering the foundation for calculus.
• Mathematically, if \(\lim_{x \to c} f(x) = L\), it says that as \(x\) gets arbitrarily close to \(c\), \(f(x)\) approaches \(L\).

Continuity

Continuity is a property of a function that indicates it doesn't have any abrupt changes, jumps, or holes. A function is continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point.
In terms of epsilon-delta, a function \(f(x)\) is continuous at a point \(c\) if for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that for every \(x\) within \(\delta\) of \(c\), \(|f(x) - f(c)| < \varepsilon\).

• If you can draw a function without lifting the pencil from the paper, it's continuous.
• Epsilon-delta definition helps prove continuity formally.
• It's essential for the function to be continuous to apply many theorems in calculus.

Calculus Proofs

Calculus proofs often involve demonstrating that a specific property holds for all allowable substitutions for variables. The epsilon-delta proof is a standard method to rigorously prove limits and thus continuity.
In our exercise, the goal was to rigorously show that \(\lim_{x \to 1} \sqrt{2x} = \sqrt{2}\) using such a proof.
The steps involved utilizing the exact definition of limits through manipulating inequalities and choosing suitable values for \(\delta\) to ensure this relationship holds.

• Such proofs reinforce the understanding of limits and continuity.
• They provide a step-by-step logical argument confirming intuitive results.
• Mastering them builds a solid foundation for further study in calculus and higher mathematics.

Real Analysis

Real analysis is a branch of mathematics focused on real numbers and the analytic properties of functions of real numbers. It's the theoretical groundwork underlying calculus.
This field rigorously proves intuitively clear concepts like limits, continuity, and differentiability, often using concepts such as epsilon-delta proofs.
In real analysis, we aim not only to understand these properties but to prove them with utmost precision and logical rigor. The exercise of proving limits with epsilon-delta is a classic example of real analysis in action.

• Real analysis distinguishes between formal and intuitive understanding.
• It requires precise definitions and proofs.
• It's essential for advanced studies in mathematical analysis and related fields.