Find its horizontal, vertical, and slant asymptote if it has any of g(x)=(x^4+1)/(x^3+x^2)

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We have to find the horizontal, vertical and slant asymptotes of g(x)=(x^4+1)/(x^3+x^2).

The function does not have any horizontal asymptote as the degree of the numerator is larger than that of the denominator.

The vertical asymptotes are the roots of the denominator.

x^3 + x^2 = 0

=> x^2(x + 1) = 0

=> x = 0 and x = -1

The vertical asymptotes are x = 0 and x = -1.

The slant asymptote is found from the quotient of x^4 + 1/x^3 + x^2

x^3 + x^2 | x^4 + 1........| x

....................x^4 + x^3


...............................1 - x^3

The slant asymptote is y = x

There is no horizontal asymptote, the vertical asymptotes are x = 0 and x = -1 and the slant asymptote is y= x.

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