Problem 1 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Question: Explain what $$\lim _{(x, y) \rightarrow(a, b)} f(x, y)=L$$ means and provide its geometric interpretation. Answer: The notation $$\lim _{(x, y) \rightarrow(a, b)} f(x, y)=L$$ represents the limit of the multivariable function f(x, y) as the point (x, y) approaches (a, b). It means that as the point (x, y) gets arbitrarily close to (a, b), the value of f(x, y) approaches L. Geometrically, f(x, y) can be visualized as a surface in 3D space, and as (x, y) approaches (a, b) in the 2D plane, the corresponding point on the surface (x, y, f(x, y)) approaches (a, b, L) in the 3D space, which means that the surface of the function approaches the height L. The limit exists only if f(x, y) gets arbitrarily close to L from all possible paths as (x, y) approaches (a, b).

Step 1: Define a Limit for a Multivariable Function

First, let's define what a limit for a function of two variables is. Suppose we have a function f(x, y) defined over a region that contains the point (a, b), except possibly at (a, b) itself. 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 small positive number ε, there exists a positive number δ such that, if $$0<\sqrt{(x-a)^2+(y-b)^2}<\delta$$ then $$|f(x, y)-L|<\epsilon$$ It means that as the point (x, y) approaches (a, b), the value of the function f(x, y) gets arbitrarily close to L.

Step 2: Geometric Interpretation

The geometric interpretation of the limit is to visualize f(x, y) as a surface in 3D space. Now, as the point (x, y) approaches (a, b) in the 2D plane, the corresponding point on the surface (x, y, f(x, y)) approaches (a, b, L) in the 3D space. So, $$\lim _{(x, y) \rightarrow(a, b)} f(x, y)=L$$ means that the surface of function f approaches the height L as the point (x, y) approaches the point (a, b). Remember that the existence of a limit depends on f(x, y) getting arbitrarily close to L from all possible paths as (x, y) approaches (a, b). If there is any path where the function does not approach L, then the limit does not exist.

Key Concepts

Limit of a Function

A limit of a function in calculus is the value that a function approaches as the input approaches a certain point. This is a fundamental concept for understanding the behavior of functions and their continuity. In the context of a multivariable function, the notation \(\lim_{(x, y) \to (a, b)} f(x, y) = L\) signifies that as the point \((x, y)\) gets closer to \((a, b)\), the function \(f(x, y)\) tends to a particular value \(L\).

• The limit \(L\) is approached as \(x\) and \(y\) come arbitrarily close to \(a\) and \(b\).
• All paths towards the limit point must be considered in multivariable limits.

This means the limit exists only if \(f(x, y)\) gets closer to \(L\) irrespective of the path taken towards \((a, b)\). If any path leads to a different value, the limit doesn't exist.

Multivariable Functions

Multivariable functions involve more than one input, typically represented by variables like \(x\) and \(y\). Such functions can be visualized in higher dimensions where each input point \((x, y)\) corresponds to a single output value \(f(x, y)\).

• Two-dimensional plane: The inputs \((x, y)\) lie on a plane.
• Three-dimensional space: The function creates a surface in \(3D\), where the height is the output \(f(x, y)\).

Understanding these functions often requires considering different paths or approaches for \(x\) and \(y\) moving towards a target point. Since variables can approach a point from various directions, multivariable functions show richer behavior than single-variable ones.

Epsilon-delta Definition

The epsilon-delta definition forms the rigorous foundation for understanding limits in both single and multivariable calculus. It provides a precise way to express how a function's value approaches a particular limit. Here's how it applies:

• For a given \(\epsilon > 0\) (an allowable difference from the limit), there is a \(\delta > 0\).
• If the distance between \((x, y)\) and \((a, b)\) is less than \(\delta\), meaning \(0 < \sqrt{(x-a)^2+(y-b)^2} < \delta\), then the difference between \(f(x, y)\) and \(L\) is less than \(\epsilon\).
• This ensures \(f(x, y)\) can be made arbitrarily close to \(L\) by choosing points close enough to \((a, b)\).

Using this definition helps in proving that the limit of a function as it approaches a particular point is a specific value. It ensures that all paths to the point \((a, b)\) converge to the same result \(L\).