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

Nov 18, 2015

I found $y = 3$

#### Explanation:

The horizontal asymptote is a line towards which the curve, described by your function, tends to get as near as possible.

To find it you can try to see what happens to your function when $x$ becomes VERY big....and see if your functions "tends" to some kind of fixed value:
as $x$ becomes very big, say $x = 1 , 000 , 000$ you have:
$f \left(1 , 000 , 000\right) = \frac{3 \cdot 1 , 000 , 000}{1 , 000 , 000 + 4}$
let us forget the $4$ that is negligible compared to $1 , 000 , 000$; you have:

$f \left(1 , 000 , 000\right) = \frac{3 \cdot \cancel{1 , 000 , 000}}{\cancel{1 , 000 , 000}} = 3$

So when $x$ becomes very big positively (and negatively, you can try this) your functions "tends" to get near the value $3$!
So the horizontal line of equation $y = 3$ will be your asymptote!

You can plot your function and see this tendency!
graph{(3x)/(x+4) [-41.1, 41.07, -20.56, 20.53]}