How to find the asymptotes of #y = x /(x-3)# ?

1 Answer
Dec 20, 2015

Answer:

I found:
#color(red)("Vertical Asymptote: "x=3)#
#color(blue)("Horizontal Asymptote: "y=1)#

Explanation:

Looking at your function you can "see" a vertical asymptote,i.e., a vertical line towards which the graph of your function will tend to get near as much as possible.

You need to look for possible prohibited #x# values and there you'll find your asymptote.

In this case the prohibited #x# value is the one that makes your function a division by ZERO: i.e., #x=3# (you cannot use #x=3# into your function so the vertical line passing through it will be a forbidden place or your asymptote).

#color(red)("Vertical Asymptote: "x=3)#

You can also "see" a horizontal asymptote,i.e., a horizontal line towards which the graph of your function will tend to.

This is a little more difficult but you can think big and imagine what happens to the graph of your function when #x# becomes VERY large and ask yourself: "it tends towards something?".

Let us take a #x# very big, say, #x=1,000,000#
and evaluate your function there so you get:

#f(1,000,000)=(1,000,000)/(1,000,000-3)~~1#

because #3#, compared to #1,000,000#, is negligible.

So you get that the horizontal line passing through: #y=f(1,000,000)=1#
will be the line towards which the graph tends to get near as much as possible:

#color(blue)("Horizontal Asymptote: "y=1)#

You can see graphically these two tendencies:
graph{x/(x-3) [-8.89, 8.89, -4.444, 4.445]}