Discussion Overview
The discussion revolves around the limit of the function $$f(x,y)=\frac{\ln(1+x^2y^2)}{x^4+y^4}$$ as $$x^2+y^2$$ approaches infinity. Participants explore different methods to analyze the limit, including polar coordinates and L'Hôpital's rule.
Discussion Character
- Exploratory
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant suggests converting to polar form to analyze the limit, proposing that the limit approaches zero as the denominator grows faster than the numerator.
- Another participant recommends using L'Hôpital's rule, indicating that multiple applications may be necessary to evaluate the limit.
- A later reply mentions that after applying L'Hôpital's rule, the numerator simplifies to $$24\cos^2(\theta)\sin^2(\theta)$$, while the denominator tends to infinity, suggesting the limit is zero.
- There is a question about the validity of stating that $$r^2 \to \infty$$ is equivalent to $$r \to \infty$$ when calculating limits, with some participants expressing that it is unusual but not incorrect.
- An alternative approach is proposed using the upper bound $$\ln(1+u) \le \sqrt{u}$$ to estimate the limit.
- Participants discuss the number of applications of L'Hôpital's rule needed, with differing opinions on whether four applications are necessary.
Areas of Agreement / Disagreement
Participants express differing views on the methods for evaluating the limit, particularly regarding the application of L'Hôpital's rule and the equivalence of limits as $$r^2$$ and $$r$$ approach infinity. No consensus is reached on the necessity of multiple applications of L'Hôpital's rule.
Contextual Notes
Participants do not fully resolve the mathematical steps involved in applying L'Hôpital's rule, and there are assumptions about the behavior of the function as $$x^2+y^2$$ approaches infinity that remain unexamined.