# How do you find the vertical, horizontal or slant asymptotes for #f(x) = (x^3-8)/(x^2-5x+6)#?

Vertical asymptotes:

Horizontal asymptotes: None

Slant asymptotes:

Vertical asymptotes:

In this case:

Horizontal asymptotes

In this function, this doesn't occur, so there are no horizontal asymptotes.

Slant asymptotes

Slant asymptotes occur when the polynomial of the denominator of a rational function has a lower degree than the polynomial of the numerator. In order to find our slant asymptote, we must divide the numerator by the denominator.

And here's your graph plotted on Desmos. (Although it seems like the graph crosses the first horizontal asymptote, the graph is actually undefined for that part.)

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To find the vertical asymptotes, you need to find the values of ( x ) that make the denominator of the function equal to zero. Then, check if these values make the numerator nonzero. For horizontal asymptotes, compare the degrees of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. If the degrees are equal, divide the leading coefficients. For slant asymptotes, perform polynomial long division and analyze the remainder. If the remainder is a nonzero polynomial, there is a slant asymptote.

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