Problem 25 - The Calculus with Analytic Geometry

Question

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

Step-by-Step Solution

Answer

The limit exists and is equal to 0.

Step 1 - Identify the limit

We need to determine if the limit \(\lim _{(x, y) \rightarrow(0,0)} \frac{x^{2} y^{2}}{x^{2}+y^{2}}\) exists.

Step 2 - Consider different paths

To determine if the limit exists, we need to check the behavior of the function along different paths approaching (0,0). Consider the path along x-axis where y=0 and along y-axis where x=0.

Step 3 - Evaluate limit along x-axis

Along the x-axis, let y=0: \(\lim_{x \to 0} \frac{x^{2} \cdot 0^{2}}{x^{2} + 0^{2}} = \lim_{x \to 0} \frac{0}{x^{2}} = 0\)

Step 4 - Evaluate limit along y-axis

Along the y-axis, let x=0: \(\lim_{y \to 0} \frac{0^{2} \cdot y^{2}}{0^{2} + y^{2}} = \lim_{y \to 0} \frac{0}{y^{2}} = 0\)

Step 5 - Evaluate limit along y=mx path

Consider another path y=mx where m is a constant: \- substitute y=mx into the function: \(\lim_{x \to 0} \frac{x^{2} (mx)^{2}}{x^{2} + (mx)^{2}} = \lim_{x \to 0} \frac{m^{2} x^{4}}{x^{2} + m^{2} x^{2}} = \lim_{x \to 0} \frac{m^{2} x^{4}}{x^{2}(1 + m^{2})} = \lim_{x \to 0} \frac{m^{2} x^{2}}{1+m^{2}} = 0\)

Step 6 - Conclude based on the paths

Since the limit along the x-axis, y-axis, and any line y=mx are all 0, the limit exists and is equal to 0.

Key Concepts

Path Approach

The path approach is a technique used to evaluate multivariable limits by exploring different paths that lead to the same point. In simpler terms, we check how the function behaves as we approach the point of interest from different directions.

When using the path approach, it's common to analyze:

• The x-axis (setting y = 0)
• The y-axis (setting x = 0)
• A linear path like y = mx, where m is a constant

By comparing these different paths, we can determine if the limit is consistent. If the limit is the same along all paths, it exists. However, if the limit varies along different paths, the limit does not exist.

For example, in the exercise, we evaluated the limit approaching (0,0) along the x-axis, y-axis, and any line y=mx, and found that all paths lead to a limit of 0. Therefore, the limit exists and is 0.

Evaluating Limits

Evaluating limits in multivariable calculus involves analyzing how a function behaves as its input values approach a specific point from different directions.

The process typically involves the following steps:

• Identify the limit
• Consider different paths approaching the point
• Evaluate the function along those paths
• Conclude based on the behavior along the paths

In the example problem, we focused on \( \frac{x^2 y^2}{x^2 + y^2} \) and checked its behavior as (x, y) approaches (0,0).
By substituting y = 0, and x = 0, we found the limit along the x-axis and y-axis respectively.
Furthermore, using the path y = mx, we substituted and simplified the function, demonstrating that all these paths result in the limit being 0.
If limits along different paths match, the overall limit exists. If they don't, it doesn't.

Multivariable Calculus

Multivariable calculus extends the concepts of single-variable calculus to functions of several variables. This branch of mathematics deals with functions that have more than one input, such as f(x, y) or f(x, y, z).

Key concepts include:

• Partial derivatives
• Multiple integrals
• Gradient, divergence, and curl

Among these concepts, evaluating limits in multiple dimensions is crucial. By understanding how a function behaves as it approaches a particular point from various directions, students can determine continuity, differentiability, and other properties.

In the exercise, by examining different paths in a two-dimensional plane, we demonstrated one of the key skills in multivariable calculus: the ability to scrutinize and identify consistent behavior across different pathways.
Mastery of these techniques enables students to tackle more complex problems involving surfaces and volumes in higher dimensions, a foundational aspect of applied mathematics, physics, and engineering.