What is the Slant Asymptote of $\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$?

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Discussion Overview

The discussion revolves around finding the slant asymptote of the rational function $\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$. The context involves calculus concepts, specifically relating to asymptotic behavior as $x$ approaches infinity.

Discussion Character

  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant mentions using long division to find the slant asymptote, resulting in $4x + 2 + \dfrac{-5x-1}{x^2-3x}$.
  • This participant notes that for very large $x$, the fraction becomes negligible, suggesting that the graph will be closer to $4x + 2$.
  • A desmos graph is referenced, indicating that $y = x + 3$ appears to be close to the slant asymptote, although this is not confirmed as the final answer.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the exact slant asymptote, as there are differing interpretations of the long division result and its implications for large $x$.

Contextual Notes

The discussion does not resolve the implications of the long division results fully, and there may be assumptions regarding the behavior of the function at infinity that are not explicitly stated.

karush
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$\tiny{s8.3.6.46}$

Find the Slant asymptote
$y=\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$

Ok the last time I did a slant asymptote was decades ago in Algrebra but this is a calculus problem

the example started with this $\displaystyle\lim_{x \to \infty}[f(x)-(mx+b)]=0$

long division returns $4x+2+\dfrac{-5x-1}{x^2-3x}$ and a desmos graph looks like y=x+3 is sort of close to the SA

so far.. anyway.. but next?
 
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karush said:
$\tiny{s8.3.6.46}$

Find the Slant asymptote
$y=\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$

Ok the last time I did a slant asymptote was decades ago in Algrebra but this is a calculus problem

the example started with this $\displaystyle\lim_{x \to \infty}[f(x)-(mx+b)]=0$

long division returns $4x+2+\dfrac{-5x-1}{x^2-3x}$ and a desmos graph looks like y=x+3 is sort of close to the SA
For very large x, that fraction will be very small so I would think the graph would be much closer to $4x+ 2$!

so far.. anyway.. but next?
 
Last edited:
i should of seen that🙄
 
https://dl.orangedox.com/GXEVNm73NxaGC9F7Cy
38 pages of calculus
50k views
 

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