Problem 24 - The Calculus with Analytic Geometry

Question

In Exercises 24 through 29 , determine if the indicated limit exists. \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y^{2}}{x^{4}+y^{4}}\)

Step-by-Step Solution

Answer

The limit does not exist.

Step 1: Understand the Problem

We need to determine if the limit \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y^{2}}{x^{4}+y^{4}}\) exists. This requires analyzing the behavior of the function as \(x\) and \(y\) approach zero.

Step 2: Check Along the x-axis and y-axis

First, evaluate the function along the x-axis (where \(y = 0\)) and the y-axis (where \(x = 0\)). \(\frac{x^2 \times 0^2}{x^4 + 0^4} = 0 \) and \(\frac{0^2 \times y^2}{0^4 + y^4} = 0 \). Thus, in both cases, the function equals 0.

Step 3: Check Along y = kx (for any constant k)

Let \( y = kx \), where \( k \) is a constant. Substitute \( y \) with \( kx \) in the function: \(\frac{x^2 (kx)^2}{x^4 + (kx)^4} = \frac{x^2 k^2 x^2}{x^4 + k^4 x^4} = \frac{k^2 x^4}{x^4 (1 + k^4)} = \frac{k^2}{1 + k^4} \). This value depends on \( k \), meaning it can vary, so the limit is not consistent.

Step 4: Conclude the Limit

Since the limit depends on the path taken (having multiple values depending on \( k \)), it is concluded that the limit does not exist.

Key Concepts

Multivariable Limits

In multivariable calculus, limits help us understand the behavior of functions at certain points, particularly when involving more than one variable. For example, consider the limit \(\frac{x^{2} y^{2}}{x^{4}+y^{4}}\) as \(x\) and \(y\) approach 0. This type of limit checks how a function behaves as we move closer to a point from multiple directions. The first step is to test the function along the basic paths, like the x-axis and y-axis. But it's critical to remember that the path doesn’t always have to be straight. Multivariable limits require checking different paths to determine if the limit is the same from all directions.

Path Dependency

Path dependency is a unique characteristic in multivariable limits. It refers to whether the value of a limit changes based on the path taken to approach the point. For instance, if we test \(\frac{x^{2} y^{2}}{x^{4}+y^{4}}\) along the x-axis and y-axis, the limit is found to be 0 for both. However, if we take a different path like \(y = kx\), where \(k\) is a constant, the function's value becomes \(\frac{k^2}{1+k^4}\). This reveals that the limit depends on the value of \( k \), which means it changes with the path. Thus, when a limit's value varies along different paths, it indicates the limit does not exist uniformly.

Limit Existence

To determine the existence of a limit in multivariable calculus, it's essential to check that it's consistent across all paths. From our example, if the limit \(\frac{x^{2} y^{2}}{x^{4}+y^{4}}\) gives different values for different paths, it proves the limit does not exist at that point. For a limit to exist, it must converge to the same value regardless of the chosen path. This principle helps confirm that a true limit is independent of the approach direction, making it a thorough test of a function’s behavior around a specific point. By understanding and testing for path dependency, we can effectively conclude whether a multivariable limit exists or not.