Question

How do you graph `f(x)=2/(x-1)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

graph{2/(x-1) [-10, 10, -5, 5]}

X intercept: Does not exist
Y intercept: (-2)

Horizontal asymptote:0
Vertical asymptote: 1

Explanation:

First of all to figure the y intercept it is merely the y value when x=0

`y=2/(0-1)`

`y=2/-1=-2`

So y is equal to `-2` so we get the co-ordinate pair (0,-2)

Next the x intercept is x value when y=0

`0=2/(x-1)`

`0(x-1)=2/`

`0=2`

This is a nonsense answer showing us that there is defined answer for this intercept showing us that their is either a hole or an asymptote as this point

To find the horizontal asymptote we are looking for when x tends to `oo` or `-oo`

`lim x to oo 2/(x-1)`

`(lim x to oo2)/(lim x to oox- lim x to oo1)`

Constants to infinity are just constants

`2/(lim x to oox-1)`

x variables to infinity are just infinity

`2/(oo-1)=2/oo=0`

Anything over infinity is zero

So we know there is a horizontal asymptote

Additionally we could tell from `1/(x-C)+D` that

C~ vertical asymptote
D~ horizontal asymptote

So this shows us that the horizontal asymptote is 0 and the vertical is 1.