# How do you find the vertical, horizontal or slant asymptotes for #y = x/(x-6)#?

vertical asymptote x = 6

horizontal asymptote y = 1

Vertical asymptotes occur as the denominator of a rational function tends to zero. To find the equation set the denominator equal to zero.

solve: x - 6 = 0 → x = 6 , is the asymptote

divide terms on numerator/denominator by x

Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here hence there are no slant asymptotes. graph{x/(x-6) [-20, 20, -10, 10]}

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To find the vertical asymptote, set the denominator equal to zero and solve for x. In this case, ( x - 6 = 0 ), so ( x = 6 ) is the vertical asymptote.

To find the horizontal asymptote, examine the degrees of the numerator and denominator. Since they have the same degree (both are linear), divide the leading coefficient of the numerator by the leading coefficient of the denominator. In this case, ( 1/1 = 1 ), so the horizontal asymptote is ( y = 1 ).

For a slant asymptote, you perform long division of the numerator by the denominator. In this case, the division gives ( y = 1 + \frac{6}{x-6} ), which means the slant asymptote is ( y = x + 1 ).

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