how to find the horizontal asymtotes of a function??

I am assuming that you are trying to graph a rational function;

    that is, a function of the type: f(x) = P(x) / Q(x).

Case 1: If the degree of P(x) is less than the degree of Q(x) then the graph as the x-axis as a horizontal                                asymptote.

Case 2: If the degree of P(x) is equal to the degree of Q(x), then the graph has an horizontal asymptote                   which is found by the equation;     y  =  a / b    where a is the leading coefficient of P(x)                                                                                                     and b is the leading coefficient of Q(x).

The degree of a polynomial is the highest exponent of the variable.

Example of case 1: f(x) = (3x + 9) / (4x² - 14)

                                        here the degree of P(x) = 1 and the degree of Q(x) = 2.

Example of case 2: f(x) = (7x³ -4x + 9) / (-x³ + x²)  ---> asymptote is y = 7 / -1   --->   y = -7

In other cases, there will be no horizontal asymptote.

Horizontal asymtotes are just the values that y cannot be but y can be some value infinitely close by.

You can look at the equation to see this.

ex 

$$\\y+3=\frac{2}{x+1}\\\\ $I see straight off that $ \frac{2}{x+1}\ne 0\\\\ $therefore $ y+3\ne0 $ therefore $y\ne-3 \\\\ $y can be any value close to 3 but it cannot be 3. $ \\ $Therefore $ y=-3 $ is a horizontal asymptote.$$ $