Discussion Overview
The discussion centers around evaluating the limit $$\lim_{x \to 1}\frac{\sin(x-1)}{x^2+x-2}$$ without using L'Hôpital's rule. Participants explore various methods of factorization and limit evaluation techniques, including the application of the limit property of sine.
Discussion Character
- Mathematical reasoning
- Exploratory
- Homework-related
Main Points Raised
- One participant initially suggests using L'Hôpital's rule but seeks alternative methods.
- Another participant recommends factorizing the denominator and using the limit property $$\lim_{x\to 0 }\frac{\sin x}{x}=1$$.
- There is a discussion about the correct factorization of the denominator, with a focus on identifying $(x-1)(x+2)$.
- Participants discuss the importance of eliminating terms that cause indeterminate forms, specifically the term $(x-1)$.
- One participant expresses confusion about how to apply the limit property after factorization, noting that the limit approaches 1 but the denominator does not approach 0.
- Several participants confirm that after proper factorization and simplification, the limit evaluates to $$\frac{1}{3}$$.
- There is a suggestion to use substitution, letting $$x-1=t$$, to further clarify the limit evaluation process.
- One participant questions the validity of canceling $$\sin(x-1)$$ and $$x-1$$ in the limit expression, seeking clarification on the reasoning behind this step.
- Another participant explains that while cancellation is permissible, it should be noted that it is based on the limit property rather than traditional fraction reduction.
Areas of Agreement / Disagreement
Participants generally agree on the final evaluation of the limit as $$\frac{1}{3}$$, but there are differing views on the methods of factorization and the appropriateness of cancellation in limit evaluation.
Contextual Notes
Some participants express uncertainty about the steps involved in factorization and limit evaluation, indicating that there may be missing assumptions or misunderstandings regarding the application of limit properties.