How do you find the horizontal asymptote for #(4x)/(x-3) #?

1 Answer
Dec 1, 2015

Answer:

I found #y=4#

Explanation:

The horizontal asymptote is a horizontal line of equation: #y="constant"#
towards which the curve described by your function TENDS to get closer and closer maybe not immediately but as #x# becomes sufficently big.

To find this line there is a trick!

Take your function and try to "see" its behavior very far from the origin...i.e. when #x# becomes VEEEEERY big!
In your case consider a #x# value very big, say, #x=1,000,000#:

you get:
#y=4*(1,000,000)/(1,000,000-3)~~4*(1,000,000)/(1,000,000)=# the #3# is negligible;
#y=4*(cancel(1,000,000))/(cancel(1,000,000))#
So, you get #y=4# that is the equation of a horizontal line that your function tends to become for #x# VEEEERY large!!
Your asymptote!!!

You can "see" this graphically:
graph{(4x)/(x-3) [-25.66, 25.65, -12.83, 12.83]}

The two branches of your function will get as near as possible to the horizontal line #y=4#!

Hope it helps!