How do you find vertical, horizontal and oblique asymptotes for #y =( x^2-x-6) /( x-2)#?

graph{[-10, 10, -10, 10]}/(x-2) (x^2-x-6)

To find the vertical asymptotes of the function (y = \frac{x^2 - x - 6}{x - 2}), we need to find the values of (x) that make the denominator (x - 2) equal to zero. The vertical asymptotes occur where the function is undefined, which happens when the denominator equals zero. Thus, the vertical asymptote is at (x = 2).

To find the horizontal asymptote, we examine the behavior of the function as (x) approaches positive and negative infinity. We can do this by dividing the leading terms of the numerator and denominator. In this case, the leading terms are both (x). So, the horizontal asymptote is (y = \frac{x^2}{x} = x).

To find the oblique asymptote, we perform polynomial long division or synthetic division on the numerator (x^2 - x - 6) divided by the denominator (x - 2). The quotient will give us the equation of the oblique asymptote.

Using long division or synthetic division:

[ \begin{array}{r|rr} x^2 - x - 6 & x & -3 \ \hline x - 2 & x^2 & -x \ & -(x^2 - 2x) & \ \hline & & x - 6 \ \end{array} ]

So, the quotient is (x - 3). Therefore, the oblique asymptote is (y = x - 3).