Problem 83 - Calculus Early Transcendentals

Question

Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)}(x+y)=a+b . \text { (Hint: Take } \delta=\varepsilon / 2 \text { ) }$$

Step-by-Step Solution

Answer

Question: Using the formal definition of limit in a two-variable system, prove that the limit of the function (x+y) as (x, y) → (a, b) is equal to (a+b). Hint: Take δ = ε/2. Answer: Our proof shows that when we consider the distance between points (x, y) and (a, b) and apply the triangle inequality theorem, we can find that for a given ε, a δ = ε/2 exists, satisfying the formal definition of limit in a two-variable system. We have shown that for any ε > 0, there exists δ > 0 such that if 0 < √((x-a)² + (y-b)²) < δ, then |(x+y)-(a+b)| < ε. Thus, we have proven that the limit of the function (x+y) as (x, y) → (a, b) is equal to (a+b).

Step 1: Introduce given exercise statement

We are given with the limit: $$\lim _{(x, y)\rightarrow(a, b)}(x+y)=a+b$$ and we are asked to prove this using the formal definition of a limit. Recall that the formal definition of limit in a two-variable system is: $$\forall \epsilon >0 \, \exists \delta > 0 : 0 < \sqrt{(x-a)^{2}+(y-b)^{2}}<\delta \Rightarrow |f(x, y)-L|<\epsilon$$ In this case, the function \(f(x, y) = x+y\) and the limit \(L = a+b\).

Step 2: Calculate distance between (x, y) and (a, b)

The first step using the formal definition of a limit is to calculate the distance between the point \((x, y)\) and \((a,b)\). The distance is given by the expression: $$0<\sqrt{(x-a)^2 + (y-b)^2} < \delta$$ We don't need to simplify it further since it already shows the distance between the two points.

Step 3: Apply the triangle inequality theorem

Now, we need to calculate \(|f(x, y) - L|\). In this case, \(f(x, y) = x+y\) and \(L = a+b\). So, we get: $$|f(x, y) - L| = |(x+y) - (a+b)| = |(x-a) + (y-b)|$$ Apply the triangle inequality theorem to the above expression: $$|(x-a) + (y-b)| \leq |(x-a)| + |(y-b)|$$

Step 4: Use the given hint δ = ε/2

The hint provided is to take \(\delta = \varepsilon / 2\). Let's now find an upper bound for the expression \(|(x-a)| + |(y-b)|\) using the given hint: $$\sqrt{(x-a)^2} \leq \sqrt{(x-a)^2 + (y-b)^2} < \delta \Rightarrow |(x-a)| < \delta = \frac{\varepsilon}{2}$$ Similarly: $$\sqrt{(y-b)^2} \leq \sqrt{(x-a)^2 + (y-b)^2} < \delta \Rightarrow |(y-b)| < \delta = \frac{\varepsilon}{2}$$

Step 5: Combine the inequalities

Combining the inequalities obtained in Step 3 and Step 4: $$|(x-a) + (y-b)| \leq |(x-a)| + |(y-b)| < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon$$

Step 6: Conclude the proof

Our goal was to prove that for any \(\varepsilon > 0\), there exists \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta\), then \(|(x+y)-(a+b)| < \varepsilon\). With our calculation, we ended up with the inequality: $$|(x-a) + (y-b)| < \varepsilon$$ That proves the statement and shows that $$\lim _{(x, y)\rightarrow(a, b)}(x+y)=a+b$$

Key Concepts

Formal Definition of a Limit

The formal definition of a limit is a fundamental concept in calculus. It helps us rigorously understand how a function behaves as its variables approach certain points. This definition sets the foundation for precise mathematical analysis. The idea is that a function approaches a specific value, called the limit, as the input variables get closer to specific values.

In multivariable calculus, the formal definition is slightly more complex, because we deal with two or more variables. Consider a function \(f(x,y)\) and suppose it has a limit \(L\) as \((x,y)\) approaches \((a,b)\). We represent this mathematically as:

• \(\forall \epsilon > 0, \exists \delta > 0\)
• Such that, if \(0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta\)
• Then, \(|f(x,y) - L| < \epsilon\)

This means that for any small number \(\epsilon\), we can find a \(\delta\) such that when the distance between \((x,y)\) and \((a,b)\) is less than \(\delta\), the difference between \(f(x,y)\) and \(L\) is less than \(\epsilon\).

Triangle Inequality Theorem

The triangle inequality theorem is a mathematical rule that applies to the distances between points. It states that in any triangle, the sum of the lengths of any two sides is greater than or equal to the length of the remaining side.

In calculus, this theorem helps us estimate bounds and simplify expressions. Given any real numbers \(x\) and \(y\), the triangle inequality theorem can be expressed as:

• \(|x + y| \leq |x| + |y|\)

When working with limits, especially in two or more variables, the triangle inequality assists in dealing with expressions that involve addition. By breaking down and bounding complex expressions like \(|(x-a) + (y-b)|\) using the theorem, we simplify calculations. This simplification is essential in formally proving limits and ensuring rigor in our conclusions.

Multivariable Calculus

Multivariable calculus is an extension of traditional calculus to functions of several variables. Instead of dealing with functions of a single variable like \(f(x)\), we work with functions of the form \(f(x, y)\), \(f(x, y, z)\), and so on. In multivariable calculus, we study aspects like partial derivatives, gradients, and the behavior of functions as vectors or points approach other vectors or points.

Understanding how limits work in this context requires looking at the function directionally — from all possible paths towards the limit point. Calculating limits in multivariable scenarios often involves considering neighborhoods around a point and requires techniques such as substitution or the squeeze theorem.
Multivariable calculus forms the backbone for many applications in physics, engineering, and data science, where variables often exist in higher-dimensional spaces.

Epsilon-Delta Definition

The epsilon-delta definition is the cornerstone of rigor in calculus. It formalizes what it means for a limit to exist. If you ever hear about making things "as small as you want," that's what epsilon and delta do!

Epsilon (\(\epsilon\)) represents how close we want the function's output to be to the limit, and delta (\(\delta\)) represents how close the input needs to be to a certain point to achieve that output precision. Here is the definition:

• \(\forall \epsilon > 0, \exists \delta > 0\)
• Such that, if \(0 < |x-a| < \delta\)
• Then, \(|f(x) - L| < \epsilon\)

In the multivariable case, we extend this concept to involve multiple variables, requiring \(0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta\) to imply \(|f(x, y) - L| < \epsilon\). Using epsilon-delta arguments is pivotal in constructing proofs about limits, allowing us to show beyond any doubt that a function behaves predictably as it approaches specific points.