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How do you find the slant asymptote of #y=((4x^3)+(x^2)+x+4)/((x^2)+5x)#?

Answer 1

James Allen

Slant asymptote is #y=(4x-19)#

While vertical asymptotes are given by solution of #(x^2+5x)=0# i.e. #x(x+5)=0# i.e. they are #x=0# and #x=-5#

To find slant asymptote of #y=(4x^3+x^2+x+4)/(x^2+5x)#, divide #(4x^3+x^2+x+4)# by #(x^2+5x)#, o

#4x(x^2+5x)-19(x^2+5x)+96x+4# i.e.

#y=(4x-19)+(96x+4)/(x^2+5x)#

Hence slant asymptote is #y=(4x-19)#

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Answer 2

Greyson Brown

To find the slant asymptote of the given rational function (y = \frac{{4x^3 + x^2 + x + 4}}{{x^2 + 5x}}), we need to perform polynomial long division. After dividing (4x^3 + x^2 + x + 4) by (x^2 + 5x), we get the quotient (4x - 5) and a remainder. The slant asymptote is given by the quotient, which is (4x - 5).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.