Question

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

Short answer

Based on the solution provided above, answer the following question: Prove that the limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function as the point (x, y) approaches (a, b). Answer: The limit of a constant multiplied by a function is equal to the constant multiplied by the limit of the function as the point (x, y) approaches (a, b), since we have shown that for every ε > 0, there exists a δ > 0 such that if 0 < sqrt((x-a)^2 + (y-b)^2) < δ, then |cf(x, y) - cL| < |c|ε, where L is the limit of the function f(x, y) as (x, y) approaches (a, b). This proves that the limit of cf(x, y) as (x, y) approaches (a, b) is equal to cL, which is the constant multiplied by the limit of the function f(x, y).

Step-by-step solution

Apply the definition of the limit to the right-hand side of the equation

Let's first consider the limit of f(x, y) as given in the problem: $$\lim _{(x, y) \rightarrow (a, b)} f(x, y) = L$$ According to the formal definition of the limit, for every ε > 0, there exists a δ > 0 such that if 0 < sqrt((x-a)^2 + (y-b)^2) < δ then |f(x, y) - L| < ε.

Consider the limit of cf(x, y) as (x, y) approaches (a, b)

Our goal is to show that the following limit holds: $$\lim _{(x, y) \rightarrow (a, b)} cf(x, y) = cL$$ If we would multiply the equation |f(x, y) - L| < ε by the constant c, we would get: $$|c(f(x, y) - L)| < |c|ε$$ Notice that the left-hand side can be rewritten as: $$|cf(x, y) - cL| < |c|ε$$ Now we want to show that for every ε > 0, there exists a δ > 0 such that if 0 < sqrt((x-a)^2 + (y-b)^2) < δ then |cf(x, y) - cL| < |c|ε.

A δ exists for the limit of cf(x, y)

Since from the assumption, for every ε > 0, there exists a δ > 0 such that if 0 < sqrt((x-a)^2 + (y-b)^2) < δ then |f(x, y) - L| < ε, we already know there exists a δ related to ε for f(x, y). We want to show that this same δ works for the limit of cf(x, y). Taking our inequality |cf(x, y) - cL| < |c|ε: If 0 < sqrt((x-a)^2 + (y-b)^2) < δ, then the related point (x, y) is within the δ-neighborhood of (a, b). If the point (x, y) is within the same neighborhood of (a, b) as related to f(x, y), then our inequality holds because we have: $$|cf(x, y) - cL| < |c|ε$$

Conclude the proof

We have shown that for every ε > 0, there exists a δ > 0 such that if 0 < sqrt((x-a)^2 + (y-b)^2) < δ, then |cf(x, y) - cL| < |c|ε. This is the definition of the limit for a function of two variables. Therefore, we have proven that: $$\lim _{(x, y) \rightarrow (a, b)} cf(x, y) = c\lim _{(x, y) \rightarrow (a, b)} f(x, y)$$

Key concepts

Formal Definition of Limit

The formal definition of a limit is a fundamental concept in calculus that helps us understand how a function behaves as it approaches a certain point. It is particularly useful in cases dealing with multivariable functions. This definition states that a function \( f(x, y) \) approaches the limit \( L \) as \( (x, y) \to (a, b) \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < \text{sqrt}((x-a)^2 + (y-b)^2) < \delta \), the inequality \( |f(x, y) - L| < \epsilon \) holds.
This means that no matter how small \( \epsilon \) is, you can always find a region within a distance \( \delta \) from \( (a, b) \) such that all points within this circle will map to \( f(x, y) \) values that are within \( \epsilon \) of \( L \). It builds the foundation for understanding limits in multivariable calculus and is an extension of the single-variable limit definition.

• Ensures functionality that gets arbitrarily close to a value \( L \)
• Important in verifying if a multivariable function reaches a limit at a point

Limits of Functions of Two Variables

When working with functions of two variables, understanding how limits operate becomes slightly more complex than with single-variable functions. Instead of a single path to approach the limit point, we have infinitely many paths that can converge on \( (a, b) \). Thus, we require the function's value to be consistent across all directions.
For a limit \( \lim_{(x, y) \to (a, b)} f(x, y) = L \) to hold, we need to show that no matter the path along which \( (x, y) \) approaches \( (a, b) \), the function value approaches \( L \). This can be plotted as a surface in 3D space, where the behavior of approaching \( L \) must be identical irrespective of the entry point or direction.

• Important for functions defined on higher-dimensional spaces
• The convergence must hold true across all possible paths

Epsilon-Delta Definition

The epsilon-delta definition is crucial for establishing the rigorous understanding of limits in calculus. It sets a precise framework for proving that a function converges to a particular limit. This formalism uses two key components, \( \epsilon \) (epsilon) and \( \delta \) (delta):
- \( \epsilon \) represents how close the output of the function must be to the expected limit \( L \).
- \( \delta \) determines the closeness of the input variable \( (x, y) \) to the point \( (a, b) \).
To prove a limit using this method, we must demonstrate that for every \( \epsilon > 0 \), there's a corresponding \( \delta > 0 \) such that all points within a radius \( \delta \) from \( (a, b) \) results in \( f(x, y) \) values within \( \epsilon \) of \( L \).
This approach is used to solidify intuitive concepts into clear mathematical proofs and ensures that calculated limits hold across any specified proximity. It involves:

• Finding a suitable \( \delta \) for each \( \epsilon \)
• Verifying the function remains close to the limit \( L \)