Math Problem Statement
Solution
The image presents three functions of two variables , each defined piecewise. Here's a summary of the three functions:
1) First function:
\begin{cases} \frac{x^3 - y^3}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases}$$ ### 2) Second function: $$f(x, y) = \begin{cases} \frac{2xy}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases}$$ ### 3) Third function: $$f(x, y) = \begin{cases} \frac{2x^2 y}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0) \\ 0 & \text{if } (x, y) = (0, 0) \end{cases}$$ The functions are discontinuous at the point $$(0, 0)$$, and the piecewise definition assigns a value of 0 when $$(x, y) = (0, 0)$$. Would you like to analyze the limits of these functions as $$(x, y)$$ approaches $$(0, 0)$$, or perhaps explore continuity and differentiability? Let me know how you'd like to proceed! Here are five related questions you might find interesting: 1. What is the limit of each function as $$(x, y) \to (0, 0)$$? 2. Are any of the given functions continuous at $$(0, 0)$$? 3. What happens if we approach $$(0, 0)$$ along different paths? 4. Can these functions be extended to make them continuous at $$(0, 0)$$? 5. What are the partial derivatives of these functions? **Tip:** When dealing with multivariable limits, it's often useful to approach the point along different paths (e.g., along $$y = 0$$, $$x = 0$$, or $$y = x$$) to check for consistency in the limit value.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Multivariable Calculus
Limits
Continuity
Differentiability
Formulas
f(x, y) = \frac{x^3 - y^3}{x^2 + y^2} for (x, y) ≠ (0, 0)
f(x, y) = \frac{2xy}{x^2 + y^2} for (x, y) ≠ (0, 0)
f(x, y) = \frac{2x^2 y}{x^2 + y^2} for (x, y) ≠ (0, 0)
Theorems
Multivariable Limit Theorem
Path Independence for Limits
Continuity at a Point in Multivariable Functions
Suitable Grade Level
University Level (Calculus III or Multivariable Calculus)
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