# How do you find the Vertical, Horizontal, and Oblique Asymptote given f(x)= (x+5)/(x+3)?

Jan 1, 2017

Horizontal: $\leftarrow y - 1 \rightarrow$.
Vertical : $\uparrow x = - 3 \downarrow$.
See Socratic graph, with asymptotes.

#### Explanation:

By actual division,

$y = f \left(x\right) = Q + \frac{R}{S} = 1 + \frac{2}{x + 3}$

y = Q gives parabolic, ( slant-straight) oblique or horizontal

asymptote according as Q is quadratic, linear or constant,

respectively.

x = zero(s) of S give vertical asymptote(s).

Here,

they are $y = 1 \mathmr{and} x = - 3$.

In the graph, the asymptote $x = - 3$ has gone into hiding. I am

unable to get it out. However, y = 1 is marked.

The graph is the a rectangular hyperbola (y-1)(x+3)=2, with center at

the meet of the asymptotes, $\left(- 3 , 1\right)$.

graph{(y-1)(x+3)(y-(x+5)/(x+3))=0 [-11.71, 11.71, -5.85, 5.86]}