Homework Help Overview
The discussion revolves around calculating the limit of the expression \(\lim _{x \to 0} \frac{\cos f(x) - \cos g(x)}{x^2}\), where \(f(x)\) and \(g(x)\) are differentiable at \(0\) and both equal to \(0\) at that point. Participants explore the implications of Cauchy's Mean Value Theorem and the conditions under which L'Hôpital's rule can be applied.
Discussion Character
- Exploratory, Assumption checking, Conceptual clarification
Approaches and Questions Raised
- Participants discuss the applicability of L'Hôpital's rule and question whether \(f\) and \(g\) are twice differentiable. There are attempts to express the limit in terms of derivatives and to clarify the definitions of differentiability and continuity at \(0\).
Discussion Status
The conversation is ongoing, with various interpretations of the problem being explored. Some participants have provided hints and suggestions for alternative approaches, such as using series expansions, while others emphasize the need for clarity regarding the differentiability of \(f\) and \(g\) around \(0\).
Contextual Notes
There is a notable emphasis on the technicalities of differentiability, particularly that \(f\) and \(g\) are not necessarily differentiable around \(0\), which complicates the application of certain mathematical tools like L'Hôpital's rule.
