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

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SUMMARY

The slant asymptote of the function \( y = \frac{4x^3 - 10x^2 - 11x - 1}{x^2 - 3x} \) is determined through polynomial long division. The division yields \( 4x + 2 + \frac{-5x - 1}{x^2 - 3x} \). As \( x \) approaches infinity, the term \( \frac{-5x - 1}{x^2 - 3x} \) becomes negligible, confirming that the slant asymptote is \( y = 4x + 2 \). Graphical analysis using Desmos suggests that the function approaches this asymptote closely for large values of \( x \).

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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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