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

Dec 1, 2015

I found $y = 4$

#### Explanation:

The horizontal asymptote is a horizontal line of equation: $y = \text{constant}$
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:
$y = 4 \cdot \frac{1 , 000 , 000}{1 , 000 , 000 - 3} \approx 4 \cdot \frac{1 , 000 , 000}{1 , 000 , 000} =$ the $3$ is negligible;
$y = 4 \cdot \frac{\cancel{1 , 000 , 000}}{\cancel{1 , 000 , 000}}$
So, you get $y = 4$ that is the equation of a horizontal line that your function tends to become for $x$ VEEEERY large!!
The two branches of your function will get as near as possible to the horizontal line $y = 4$!