Question

Use the formal definition of a limit to prove that $$\lim _{(x, y) \rightarrow(a, b)}(f(x, y)+g(x, y))=\lim _{(x, y) \rightarrow(a, b)} f(x, y)+$$ $$\lim _{(x, y) \rightarrow(a, b)} g(x, y)$$

Short answer

Based on the step-by-step solution provided, answer the following question: **Question:** Prove that for continuous functions \(f(x, y)\) and \(g(x, y)\), \(\lim_{(x, y) \rightarrow (a, b)} (f(x, y) + g(x, y)) = \lim_{(x, y) \rightarrow (a, b)} f(x, y) + \lim_{(x, y) \rightarrow (a, b)} g(x, y)\). **Answer:** Using the formal definition of a limit and the given continuous functions, we have shown that for any \(\epsilon > 0\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^2+(y-b)^2} < \delta\), then \(| (f(x, y) + g(x, y)) - (L_1 + L_2) | < \epsilon\), where \(L_1\) and \(L_2\) are the respective limits of the functions \(f(x, y)\) and \(g(x, y)\). This implies that \(\lim_{(x, y) \rightarrow (a, b)} (f(x, y) + g(x, y)) = L_1 + L_2 = \lim_{(x, y) \rightarrow (a, b)} f(x, y) + \lim_{(x, y) \rightarrow (a, b)} g(x, y)\), proving the desired result.

Step-by-step solution

Recap the formal definition of a limit

The formal definition of limit states: Given a function f(x,y), we say that the limit of f(x,y) as (x, y) approaches (a, b) is L, denoted as \(\lim_{(x, y) \rightarrow (a, b)} f(x, y) = L\), if for any positive \(\epsilon\), there exists a \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^2+(y-b)^2} < \delta\), then \(|f(x, y) - L| < \epsilon\).

Define the given functions and their limits

Let \(f(x, y)\) and \(g(x, y)\) be continuous functions defined in the neighborhood of (a, b) (except possibly at (a, b) itself), and let: - \(L_1 = \lim_{(x, y) \rightarrow (a, b)} f(x, y)\) - \(L_2 = \lim_{(x, y) \rightarrow (a, b)} g(x, y)\) We need to prove that \(\lim_{(x, y) \rightarrow (a, b)} (f(x, y) + g(x, y)) = L_1 + L_2\).

Use the given functions in the formal definition of limit

By the formal definition of a limit, for any \(\epsilon > 0\), we need to find a \(\delta > 0\) such that if \(0 < \sqrt{(x-a)^2+(y-b)^2} < \delta\), then \(| (f(x, y) + g(x, y)) - (L_1 + L_2) | < \epsilon\).

Break the inequality into two functions

Let's rewrite the inequality by adding and subtracting \(L_1 + g(x, y)\) inside the absolute value: \(|(f(x, y) - L_1) + (g(x, y) - L_2)| < \epsilon\) By triangle inequality, we have: \(|(f(x, y) - L_1) + (g(x, y) - L_2)| \leq |f(x, y) - L_1| + |g(x, y) - L_2|\). Now, we need to find a suitable \(\delta\) for our limit to hold.

Choose suitable \(\epsilon/2\) for each limit

Since \(L_1 = \lim_{(x, y) \rightarrow (a, b)} f(x, y)\) and \(L_2 = \lim_{(x, y) \rightarrow (a, b)} g(x, y)\), by the formal definition of a limit, for any positive \(\epsilon / 2\), we can find a \(\delta_1 > 0\) and a \(\delta_2 > 0\) such that if \(0 < \sqrt{(x-a)^2+(y-b)^2} < \delta_1\), then \(|f(x, y) - L_1| < \epsilon / 2\), and if \(0 < \sqrt{(x-a)^2+(y-b)^2} < \delta_2\), then \(|g(x, y) - L_2| < \epsilon / 2\).

Choose the smallest \(\delta\)

Now choose \(\delta = \min(\delta_1, \delta_2)\). Substituting this value of \(\delta\) will give us: \(|f(x, y) - L_1| + |g(x, y) - L_2| < \epsilon / 2 + \epsilon / 2 = \epsilon\).

Conclude the proof

Since we have shown that for any given \(\epsilon > 0\), there exists a \(\delta > 0\) such that the inequality holds, we can conclude that: $$\lim_{(x, y) \rightarrow (a, b)} (f(x, y) + g(x, y)) = L_1 + L_2 = \lim_{(x, y) \rightarrow (a, b)} f(x, y) + \lim_{(x, y) \rightarrow (a, b)} g(x, y)$$

Key concepts

Formal Definition of a Limit

The formal definition of a limit is crucial for understanding how we prove limits in calculus. The definition states that the limit of a function \( f(x, y) \) as the point \( (x, y) \) approaches \( (a, b) \) is \( L \), denoted as \( \lim_{(x, y) \rightarrow (a, b)} f(x, y) = L \). This expression means that for every \( \epsilon > 0 \), no matter how small, there exists a number \( \delta > 0 \) such that if the distance from \( (x, y) \) to \( (a, b) \) is less than \( \delta \), then the value of \( f(x, y) \) will be within \( \epsilon \) of \( L \).
This precision allows us to mathematically prove that a function approaches a certain value as the input approaches a given point. The crucial part of this definition is the "squeeze" by \( \epsilon \) and \( \delta \), which ensures that the function gets arbitrarily close to the limit \( L \).

Epsilon-Delta Definition

The epsilon-delta definition is a formal way to specify limits with great accuracy. Here's how it works: given any positive number \( \epsilon \), you choose a corresponding \( \delta > 0 \). If the distance from \( (x, y) \) to \( (a, b) \) is smaller than \( \delta \), we ensure that \( |f(x, y) - L| < \epsilon \).
This involves:

• Choosing \( \epsilon \) arbitrarily small.
• Finding a \( \delta \) such that touching the point \( (x, y) \) closer than \( \delta \) keeps the function within the bounds set by \( \epsilon \).

The challenge lies in setting \( \delta \) correctly, especially when dealing with complex functions. Once effectively chosen, it guarantees the function's value nears the limit value as closely as desired.

Sum of Limits

One interesting property of limits is the sum of limits theorem. It states that if \( \lim_{(x, y) \to (a, b)} f(x, y) = L_1 \) and \( \lim_{(x, y) \to (a, b)} g(x, y) = L_2 \), then \( \lim_{(x, y) \to (a, b)} (f(x, y) + g(x, y)) = L_1 + L_2 \).
This conclusion arises naturally from the formal definition. Because we can control how close each function's value is to its own limit individually, we're naturally able to control their sum. Essentially, when each function individually sticks close to its respective limits, their sum will stick close to the sum of these limits.
By dividing \( \epsilon \) into smaller parts like \( \epsilon/2 \), we ensure both individual convergence contribute to the overall sum, making this sum theorem a powerful tool in calculus.

Continuity of Functions

Continuity of a function at a point is directly tied to the function's limit. A function \( f(x, y) \) is continuous at point \( (a, b) \) if the limit as \( (x, y) \) approaches \( (a, b) \) equals \( f(a, b) \).
Continuity intuitively means no jumps or gaps in the graph of the function around that point. This trait assures us that as we get closer to \( (a, b) \), the function values approach a steady outcome which is \( f(a, b) \).
Beyond just ensuring close succession of points produces similar results, continuity aids in handling multiple functions together. For instance, combining or summing continuous functions inherently results in another continuous function, as long as we stay within the bounds of their defined limits. This makes continuity a key player in comprehending and applying calculus principles to complex real-world scenarios.