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

Apr 22, 2018

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

$y = \frac{2}{0 - 1}$

$y = \frac{2}{-} 1 = - 2$

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

$0 = \frac{2}{x - 1}$

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 $\infty$ or $- \infty$

$\lim x \to \infty \frac{2}{x - 1}$

$\frac{\lim x \to \infty 2}{\lim x \to \infty x - \lim x \to \infty 1}$

Constants to infinity are just constants

$\frac{2}{\lim x \to \infty x - 1}$

x variables to infinity are just infinity

$\frac{2}{\infty - 1} = \frac{2}{\infty} = 0$

Anything over infinity is zero

So we know there is a horizontal asymptote

Additionally we could tell from $\frac{1}{x - C} + D$ that

C~ vertical asymptote
D~ horizontal asymptote

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