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

1 Answer
Feb 12, 2017

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]}