Question

How do you find all horizontal and vertical asymptotes of `f(x)=arctan x- 1/(x-1)`?

Answer 1

Curvilinear asymptote : `y = arc tan x`.
Vertical asymptote: `uarr x = 1 darr`.
Horizontal asymptotes: `y = +- pi / 2.`

Explanation:

The form

`y = Q(x)+(R(x))/(P(x))`, reveals asymptotes

`y = Q(x)=arc tan x and P(x)=x-1=0`.

The first is a curvilinear asymptotes that has its outer asymptotes

`y=+-pi/2`.

See below the grandeur of the clustering, on either side of x = 1,

when general values are allowed to arc tan x. It is indeed marching

to `oo`, in the opposite directions of the common-to-all asymptote

x = 1.

Here, the horizontal asymptotes are # y = (2 k + 1 ) pi/2, k = 0, +-1,

+-2 +-3, ...#.

These are the asymptotes of the curvilinear asymptotes

`y = k pi + arc tan x, k = 0, +-1, +-2, +-3, ...`

graph{(x - tan (y + 1 / ( x - 1 )))(x - 1) = 0[-2 3 -5 5]}