Question

Prove that\(\begin{aligned} \lim _{(x, y) \rightarrow(a, b)}[f(x, y)+g(x, y)&] \\\=& \lim _{(x, y) \rightarrow(a, b)} f(x, y)+\lim _{(x, y) \rightarrow(a, b)} g(x, y) \end{aligned}\) provided that the latter two limits exist.

Short answer

The limit of the sum is the sum of the limits: \( L_f + L_g \).

Step-by-step solution

Understand the Limit Definition

The limit \( \lim _{(x, y) \to (a, b)} f(x, y) \) exists if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta \), then \( |f(x, y) - L_f| < \epsilon \), where \( L_f \) is the limit of function \( f \) as \( (x, y) \) approaches \( (a, b) \). Similar conditions apply for \( g(x, y) \) with limit \( L_g \).

Express the Sum Limit

For \( \lim _{(x, y) \to (a, b)} [f(x, y) + g(x, y)] \) to exist, \( f(x, y) + g(x, y) \) must approach a sum \( L_s \) as \( (x, y) \) approaches \( (a, b) \), specifically, \( L_s = L_f + L_g \), where \( L_f \) and \( L_g \) are the individual limits of \( f \) and \( g \), respectively.

Apply Limit Definition to Each Function

Assume \( \lim _{(x, y) \to (a, b)} f(x, y) = L_f \) and \( \lim _{(x, y) \to (a, b)} g(x, y) = L_g \), given in the problem statement. This means there are \( \delta_1 \) for \( f(x, y) \) and \( \delta_2 \) for \( g(x, y) \) such that satisfying the epsilon-delta conditions yields results close to \( L_f \) and \( L_g \).

Combine Epsilon-Delta Conditions for the Sum

Choose \( \delta = \min(\delta_1,\delta_2) \). If \( \sqrt{(x-a)^2 + (y-b)^2} < \delta \), then:1. \( |f(x, y) - L_f| < \epsilon/2 \)2. \( |g(x, y) - L_g| < \epsilon/2 \).Thus, \( |(f(x, y) + g(x, y)) - (L_f + L_g)| < |f(x, y) - L_f| + |g(x, y) - L_g| < \epsilon/2 + \epsilon/2 = \epsilon \).

Conclude the Proof

Since the conditions of the epsilon-delta definition are satisfied for \( f(x, y) + g(x, y) \), we have shown that \( \lim _{(x, y) \to (a, b)} [f(x, y) + g(x, y)] = L_f + L_g \). This confirms that if the two limits individually exist, their sum also has a limit, equal to the sum of the individual limits.

Key concepts

Limit Theorem

The Limit Theorem is a foundational principle in calculus that explains how limits interact with certain operations, such as addition or multiplication. In the context of multivariable calculus, the theorem states that if the limits of two functions, \( f(x, y) \) and \( g(x, y) \), exist as they approach a certain point \((a, b)\), then the limit of their sum also exists and is equal to the sum of their individual limits. This is succinctly expressed as:

• \( \lim_{(x, y) \to (a, b)} [f(x, y) + g(x, y)] = \lim_{(x, y) \to (a, b)} f(x, y) + \lim_{(x, y) \to (a, b)} g(x, y) \)

This theorem simplifies the analysis of complex functions by allowing us to break them down into simpler components that are easier to work with. It is applicable provided the individual limits exist.

Epsilon-Delta Definition

The epsilon-delta definition is a rigorous way to define the concept of a limit. For a function \( f \) of two variables \( (x, y) \), the limit \( \lim_{(x, y) \to (a, b)} f(x, y) = L \) exists if, for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < \sqrt{(x-a)^2 + (y-b)^2} < \delta \), it follows that \( |f(x, y) - L| < \epsilon \).

• \( \epsilon \) represents any small positive number that defines how close \( f(x, y) \) must get to \( L \).
• \( \delta \) provides a radius around the point \((a, b)\) where this closeness condition holds.

The epsilon-delta approach ensures precision and clarity in defining limits, which is crucial in more complex analyses.

Function Limits

Function limits are vital in understanding how functions behave near certain points. In multivariable calculus, limits describe the behavior of a function \( f(x, y) \) as \( (x, y) \) approaches a specific point \((a, b)\). Understanding function limits helps to:

• Determine continuity at a point.
• Analyze the behavior of surfaces and 3D shapes represented by equations.
• Lay groundwork for derivatives and integrals in higher dimensions.

The goal is to establish what value the function "leans" towards as \( (x, y) \) gets very close to \((a, b)\). This conceptualization becomes critical when solving real-world engineering and scientific problems.

Sum of Limits

The sum of limits concept reveals how the limits of function sums are handled. Based on the Limit Theorem, if \( \lim _{(x, y) \to (a, b)} f(x, y) = L_f \) and \( \lim _{(x, y) \to (a, b)} g(x, y) = L_g \) both exist, then their sum must also converge, and:

• \( \lim _{(x, y) \to (a, b)} [f(x, y) + g(x, y)] = L_f + L_g \)

This ensures that calculating the limit of complex sums can be simplified by tackling each part individually. It underscores the use of the epsilon-delta definition to guarantee that each function adheres closely to its respective limit, ensuring that their combination does the same. This process is crucial for proofs involving sums and for solving equations accurately in applied mathematics.