Question

Use the formal definition of a limit to prove that \(\lim _{(x, y) \rightarrow(a, b)} y=b,\) (Hint. Take \(\delta=\varepsilon,\) ).

Short answer

Question: Prove that the limit of a function y equals b as (x, y) approaches (a, b) using the formal definition of a limit. Answer: Using the epsilon-delta definition of a limit and taking δ = ε, it is proven that for any ε > 0, there exists a δ > 0 such that for all (x, y) in the domain of f, if \(0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta,\) then \(|y - b| < \epsilon\). This satisfies the definition of a limit, so we can conclude that \(\lim_{(x, y) \rightarrow (a, b)} y = b.\)

Step-by-step solution

Recall the Epsilon-Delta Definition of a Limit

The epsilon-delta definition of a limit states that for any given \(\epsilon > 0\), there exists a \(\delta > 0\) such that for all (x, y) in the domain of f, if \(0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta\), then \(|y - b| < \epsilon\).

Create an Inequality

Given \( 0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta \), we need to find an upper bound for \(|y - b|\) . We can begin by applying the triangle inequality to separate the terms: $$ \sqrt{(x-a)^2+(y-b)^2} \ge |y-b|.$$ Now we have an inequality that includes \(|y - b|\).

Choose an Appropriate Value for δ

According to the hint provided, we should take \(\delta = \epsilon\). So let's plug this value into the inequality we obtained in step 2: $$ |y-b|\le\sqrt{(x-a)^2+(y-b)^2} <\delta=\epsilon.$$ This shows that for any ε > 0, there exists a δ > 0 such that for all (x, y), if \(0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta,\) then \(|y - b| < \epsilon\). The inequality satisfies the epsilon-delta definition of a limit, so we've proven that \(\lim_{(x, y) \rightarrow (a, b)} y = b.\)

Key concepts

Epsilon-Delta Definition

To understand the Epsilon-Delta Definition, imagine you want to prove the limit of a function as it approaches a certain point. This definition helps establish the behavior of the function at that point without necessarily reaching it. It's like getting closer and closer to a goal without actually touching it.

The definition uses two parameters:

• Epsilon (\(\varepsilon\text{ > }0\)): Represents how close the function's value (\(|f(x,y) - L|\)) needs to be to the limit (L).
• Delta (\(\delta\text{ > }0\)): Describes how near the input values (\((x, y)\)) must be to the point of interest (\((a, b)\)).

Essentially, for every positive \(\varepsilon\), there must be a corresponding \(\delta\) ensuring the proximity of the function's output to its limit when the domain's elements are within a \(\delta\) range.

This concept can best be visualized as a shrinking zone around a limit point where within that zone, the function's value remains very close to the limit. It's a mathematical representation of precision and accuracy in evaluating limits.

Multi-variable Limits

Multi-variable limits extend the idea of limits from single-variable calculus to functions involving multiple variables. The key is to determine the behavior of these functions as their variables approach specified points in a multidimensional space.

Consider a two-variable function \(f(x, y)\). The limit as \((x, y)\) approaches \((a, b)\) is denoted as \(\lim_{(x, y) \to (a, b)} f(x, y) = L\). This means that as \(x\) and \(y\) get closer to \(a\) and \(b\), respectively, the function value \(f(x, y)\) gets closer to L.

The Epsilon-Delta Definition can still be applied here, requiring a 2D domain of the function. When applying this, visualize a small circle around \((a, b)\). If the outputs of \(f(x, y)\) fall within an epsilon neighborhood of L, as the inputs fall within a delta neighborhood of \((a, b)\), it proves the limit. This approach ensures we handle limits comprehensively, even for complex functions in multiple dimensions.

Triangle Inequality

The Triangle Inequality is a fundamental concept in mathematics that aids in understanding the properties of distances and absolute values. It states that for any vectors \(x\) and \(y\) in a space, the distance between \(x\) and \(y\) is less than or equal to the sum of their lengths individually. For instance, \(|x + y| \leq |x| + |y|\).

In the context of limit proofs, the Triangle Inequality supports breaking down more complex expressions into manageable parts. Consider its application in estimating the value \(|y - b|\). By using the inequality:
\[\sqrt{(x-a)^2 + (y-b)^2} \geq |y-b|\], it becomes easier to analyze the behavior of multi-variable functions. The inequality helps ensure that the path taken to reach the limit does not affect the result, maintaining consistency across all approaches.

By leveraging this inequality, you gain confidence that any detours or paths influence the multifaceted functions' calculations in a predictable manner. Thus, the Triangle Inequality plays a crucial role in proving and comprehending limit scenarios.