Question

How to find a horizontal asymptote `(x^2 - 5x + 6)/( x - 3)`?

Answer 1

There is no horizontal asymptote

Explanation:

There is no horizontal asymptote as degree of numerator `2` is greater than that of denominator `1` by one.

In such case, there is a possibility of a slant asymptote, but before concluding that let us factorize `(x^2−5x+6)` as follows:

`(x^2−5x+6)=x^2-3x-2x+6=x(x-3)-2(x-3)=(x-2)(x-3)`

Hence `(x^2−5x+6)/(x−3)` can be simplified as follows:

`((x-2)(x-3))/(x-3)` or

`(x-2)`

Hence `(x^2−5x+6)/(x−3)` is the equation of just the line `y=(x-2)`
(I do not think tis can be considered as slanting asymptote).

But as `x-3` appears in denominator the domain of`x` in `y` does not include `3`.