Discussion Overview
The discussion centers on evaluating and simplifying the limit of the expression \(((x^2+y^2+1)^{1/2}) - 1\) as \((x, y)\) approaches \((0, 0)\). Participants explore different methods for finding the limit, including algebraic simplification and polar coordinates.
Discussion Character
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant claims the limit equals 0 by simplifying the expression and evaluating it directly at \((0, 0)\), but questions the correctness of this approach.
- Another participant suggests that converting to polar coordinates yields a different result, referencing confirmation from Wolfram Alpha.
- A request for clarification on the polar coordinate method is made, indicating some confusion about the approach.
- A later reply outlines the polar coordinate transformation, stating that the limit can be expressed as \(\lim_{r\to0}\left(\frac{r^2}{\sqrt{r^2+1}-1} \right)\) and mentions the possibility of using L'Hôpital's Rule after rationalizing the denominator.
Areas of Agreement / Disagreement
Participants do not appear to agree on the limit's value, as different methods yield different results. The discussion remains unresolved with multiple competing views on the correct approach.
Contextual Notes
There are indications of missing assumptions regarding the methods used, particularly in the transition to polar coordinates and the application of L'Hôpital's Rule. The dependence on specific mathematical techniques may affect the outcomes.
