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

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In summary, the conversation discusses finding the slant asymptote for the function $y=\dfrac{4x^3-10x^2-11x-1}{x^2-3x}$ using the limit definition and long division. The suggested slant asymptote is $y=4x+2$, and it is supported by a desmos graph. It is also mentioned that there is a 38-page document and a video with 50k views that may provide more information on this topic.
  • #1
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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  • #2
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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  • #3
i should of seen that🙄
 
  • #4
https://dl.orangedox.com/GXEVNm73NxaGC9F7Cy
38 pages of calculus
50k views
 

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

What is a slant asymptote?

A slant asymptote is a line that a graph approaches but never touches. It occurs when the degree of the numerator of a rational function is exactly one more than the degree of the denominator.

How do you find a slant asymptote?

To find a slant asymptote, you need to divide the numerator by the denominator using long division. The quotient will be the equation of the slant asymptote.

What does the slant asymptote represent?

The slant asymptote represents the long-term behavior of the graph. It shows the direction the graph will approach as the input values get larger or smaller.

Can a graph have more than one slant asymptote?

Yes, a graph can have more than one slant asymptote if the degree of the numerator is two or more greater than the degree of the denominator. In this case, there will be multiple lines that the graph approaches but never touches.

Do all rational functions have a slant asymptote?

No, not all rational functions have a slant asymptote. If the degree of the numerator is equal to or less than the degree of the denominator, the graph will have a horizontal asymptote instead.

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