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

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The discussion focuses on finding the slant asymptote of the function \( \frac{4x^3-10x^2-11x-1}{x^2-3x} \). Long division of the polynomial yields \( 4x + 2 + \frac{-5x-1}{x^2-3x} \), indicating that the slant asymptote is approximately \( y = 4x + 2 \) for large values of \( x \). A graphing tool suggests that the line \( y = x + 3 \) is close to the asymptote, but further analysis confirms that the actual asymptote is indeed \( 4x + 2 \). The discussion highlights the importance of understanding the behavior of the fraction as \( x \) approaches infinity. Overall, the key takeaway is that the slant asymptote for the given function is \( y = 4x + 2 \).
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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?
 
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i should of seen that🙄
 
https://dl.orangedox.com/GXEVNm73NxaGC9F7Cy
38 pages of calculus
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