show all work to identify the asymptotes and zero of the function f(x)= 6x/x^2-36

Vertical asymptotes    and

Horizontal asymptote  

Zero of the function      

Step-by-step explanation:

We have the following expression

We can simplify the denominator by factoring the quadratic function.

Remember the property of difference of squares.

So:

Then:

The vertical asymptotes of the function are vertical lines that pass through the points that make the function denominator zero

Note that when x = 6 or when x = -6 the function denominator is equal to zero. But the division between zero is not defined.

herefore the function has the following vertical asymptotes:

The horizontal asymptote is calculated by the following limit

When evaluating the limit we have an indeterminacy of the type:

Then we divide each term of the function by the variable with the greatest exponent. In this case is:

                                    ↓

So the horizontal asymptote is:

The zero of the function is the point where .

You can see that when

So the zero of the function is the point: (0,0)