Discussion Overview
The discussion revolves around finding the slant asymptote of the rational function $$\frac{{x}^{3}-5{x}^{2}+4x}{-4{x}^{2}+36}$$. Participants explore methods such as polynomial long division and limits to determine the coefficients of the asymptote.
Discussion Character
- Technical explanation, Mathematical reasoning, Debate/contested
Main Points Raised
- One participant suggests that the slant asymptote can be expressed as $$mx+b$$, proposing that $$m = -\frac{1}{4}$$ but is uncertain about the value of $$b$$.
- Another participant recommends performing polynomial long division to find the asymptote, noting that the remainder will approach zero for large values of $$x$$.
- A different approach is presented involving limits to isolate $$b$$, with a reminder that for some functions, limits as $$x \to -\infty$$ may also be necessary.
- One participant expresses confusion about obtaining the correct remainder during long division, indicating a possible discrepancy in their calculations.
- Another participant provides a detailed long division process, concluding that the asymptote is $$y = -\frac{1}{4}x + \frac{5}{4}$$ and notes that the remainder will diminish for large $$x$$.
- A later reply confirms the asymptote found by the previous participant and describes a verification method for the long division result.
- One participant expresses curiosity about the LaTeX formatting used for long division.
Areas of Agreement / Disagreement
There is a general agreement on the form of the slant asymptote as $$y = -\frac{1}{4}x + \frac{5}{4}$$, but uncertainty remains regarding the calculation of $$b$$ and the correctness of the long division process among some participants.
Contextual Notes
Participants have not fully resolved the discrepancies in long division results, and there are varying approaches to determining the value of $$b$$, indicating potential limitations in their methods.