Question

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

Answer 1

Horizontal: `larr y - 1 rarr`.
Vertical : `uarr x = -3 darr`.
See Socratic graph, with asymptotes.

Explanation:

By actual division,

`y = f(x)=Q+R/S=1+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 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, `(-3, 1)`.

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