# How do you find the Vertical, Horizontal, and Oblique Asymptote given f(x)= (x^2-5x+6)/ (x^2-8x+15)?

Sep 18, 2016

vertical asymptote at x = 5
horizontal asymptote at y = 1

#### Explanation:

The first step is to factorise and simplify f(x).

$f \left(x\right) = \frac{\left(x - 2\right) \cancel{\left(x - 3\right)}}{\left(x - 5\right) \cancel{\left(x - 3\right)}} = \frac{x - 2}{x - 5}$

The denominator of f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve : $x - 5 = 0 \Rightarrow x = 5 \text{ is the asymptote}$

Horizontal asymptotes occur as

${\lim}_{x \to \pm \infty} , f \left(x\right) \to c \text{ (a constant)}$

divide terms on numerator/denominator by x

$f \left(x\right) = \frac{\frac{x}{x} - \frac{2}{x}}{\frac{x}{x} - \frac{5}{x}} = \frac{1 - \frac{2}{x}}{1 - \frac{5}{x}}$

as $x \to \pm \infty , f \left(x\right) \to \frac{1 - 0}{1 - 0}$

$\Rightarrow y = 1 \text{ is the asymptote}$

Oblique asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here (both of degree 1 ) Hence there are no oblique asymptotes.
graph{(x-2)/(x-5) [-20, 20, -10, 10]}