Math Problem Statement
Solution
To determine the set of all non-negative values of for which the function is continuous at , we need to analyze the given function:
\frac{2xy}{(x^2 + y^2)^k} & \text{if } (x,y) \neq (0,0), \\ 0 & \text{if } (x,y) = (0,0). \end{cases}$$ ### Continuity at $$(0,0)$$ For $$f$$ to be continuous at $$(0,0)$$, the limit of $$f(x,y)$$ as $$(x,y)$$ approaches $$(0,0)$$ must equal $$f(0,0)$$, which is 0. So, we need to check: $$\lim_{(x,y) \to (0,0)} \frac{2xy}{(x^2 + y^2)^k} = 0$$ ### Polar Coordinates It is useful to convert the function into polar coordinates: $$x = r \cos \theta, \quad y = r \sin \theta$$ Then, $$f(x,y) = \frac{2xy}{(x^2 + y^2)^k} = \frac{2(r \cos \theta)(r \sin \theta)}{(r^2)^k} = \frac{2r^2 \cos \theta \sin \theta}{r^{2k}} = 2 \cos \theta \sin \theta \cdot r^{2-2k}$$ ### Limit Analysis To find the limit as $$r \to 0$$: $$\lim_{r \to 0} 2 \cos \theta \sin \theta \cdot r^{2-2k}$$ For the limit to be zero, the exponent $$2-2k$$ must be positive (since $$\cos \theta \sin \theta$$ is bounded). Therefore: $$2 - 2k > 0 \implies 2 > 2k \implies k < 1$$ ### Conclusion The function $$f$$ is continuous at $$(0,0)$$ if $$k < 1$$. Thus, the set of all non-negative values of $$k$$ for which $$f$$ is continuous at $$(0,0)$$ is: $$\boxed{[0,1)}$$ Hence, the correct answer is: (B) $$\{ k \in \mathbb{R} : 0 \leq k < 1 \}$$ Do you want any more details or have any questions? Here are 8 relative questions that expand on this problem: 1. How would the continuity condition change if $$k$$ could also be negative? 2. What is the behavior of the function as $$k$$ approaches 1 from the left? 3. How do different values of $$\theta$$ affect the limit analysis in polar coordinates? 4. Can you explain the significance of converting to polar coordinates in continuity problems? 5. What would the function look like for specific values of $$k$$ within the range $$[0,1)$$? 6. How would you prove the continuity for a specific value of $$k$$ in the allowed range? 7. What happens to the function if $$k = 1$$? 8. How does the choice of the definition of $$f$$ at $$(0,0)$$ affect its continuity? **Tip:** When analyzing the continuity of functions at a point, always consider converting to polar coordinates if the function involves $$x^2 + y^2$$. This can simplify the problem significantly.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Limits
Continuity
Polar Coordinates
Formulas
Limit definition
Conversion to polar coordinates
Theorems
Epsilon-delta definition of continuity
Suitable Grade Level
Advanced High School or College