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Chapter 12: Problem 1

Explain what \(\lim _{(x, y) \rightarrow(a, b)} f(x, y)=L\) means.

Short Answer

Expert verified
Answer: The limit of a function for two variables represents the value the function approaches as we get closer to a specific point (a, b) from any direction. It is formally defined as \(\lim_{(x, y) \rightarrow (a, b)} f(x, y) = L\) if and only if for any small positive value \(\epsilon\), there exists a corresponding positive value \(\delta\) such that whenever the distance between (x, y) and (a, b) is less than \(\delta\) but greater than 0, the distance between f(x, y) and the limit L is less than \(\epsilon\). This definition ensures that the function's behavior is consistent around the point (a, b) regardless of the direction of approach.

Step by step solution

01

Introduce the concept of limit in two variables

The limit of a function for two variables, denoted as \(\lim_{(x, y) \rightarrow (a, b)} f(x, y)\), represents the value the function approaches as we get closer and closer to the point \((a, b)\) from any direction. If the limit exists, it means that the function's behavior is consistent as we approach this point from any path and the function approaches a unique value, denoted as \(L\).
02

Define the limit formally

Formally, we can say that the limit of a function of two variables \(f(x, y)\) at the point \((a, b)\) is \(L\) if and only if $$\lim_{(x, y) \rightarrow (a, b)} f(x, y) = L \Leftrightarrow \forall \epsilon > 0, \exists \delta > 0 \text{ such that } 0<\sqrt{(x - a)^2 + (y - b)^2}< \delta \Rightarrow |f(x, y) - L| < \epsilon.$$
03

Interpret the formal definition

The definition states that for any small positive value \(\epsilon\), there exists a corresponding positive value \(\delta\) such that whenever the distance between \((x, y)\) and \((a, b)\) is less than \(\delta\) but greater than \(0\), the distance between \(f(x, y)\) and the limit \(L\) is less than \(\epsilon\). This means that if we can get arbitrarily close to the point \((a, b)\), then we can also get arbitrarily close to the value \(L\). The function's behavior is consistent around the point \((a, b)\) despite the direction of approach.
04

Understand the importance of two-variable limits

The concept of limits for functions of two or more variables is foundational in multivariable calculus, where continuity, differentiation, and integration across domains of multiple variables are studied. It is an extension of single-variable limits and serves as a crucial precursor for understanding more complex topics in higher dimensions.

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Most popular questions from this chapter

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