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

Question

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

Lily Anderson · Accepted Answer

I found #y=4#

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!

Elijah Allen · Answer

undefined To find the horizontal asymptote of the function $ \frac{4x}{x-3} $, you can use the concept of limits as $ x $ approaches positive or negative infinity.

As $ x $ approaches infinity, the terms in the numerator and denominator with the highest power dominate the fraction. In this case, the highest power terms are $ 4x $ and $ x $ respectively. So, the horizontal asymptote can be found by dividing the leading coefficient of the numerator by the leading coefficient of the denominator.

Therefore, the horizontal asymptote is $ y = \frac{4}{1} = 4 $.