Question

Find the vertical and horizontal asymptote(s), if any, of the following function...

Answer 1

\[f(x)=\frac{ (6x^2+x+12) }{(3x^2−5x+2)(x^2+2) }\]

Answer 2

So very confused :/ Please help..

Answer 3

It would have a vertical asymptote at the line x = a if
\[\lim_{x \rightarrow a}f(x)=\pm \infty\]

Answer 4

Possibly for values of x where the denominator (but not the numerator) is zero.

Answer 5

Do you mind kinda working it out with me because I have no clue how to get the answer..

Answer 6

Sure :)
The denominator is
(3x^2 - 5x + 2)(x^2 + 2)
Now when is this equal to zero?

Answer 7

Thanks, and Im not sure do I have to set both of them equal to zero?

Answer 8

No...
ab = 0 means a = 0 or b = 0, only one of then need be zero
besides...
(x^2 + 2) is never zero for any real values of x. So I suggest you look at the other factor :)

Answer 9

So..? \[(3x^2 - 5x + 2)=0\]

Answer 10

Yes..., so can you work it out?

Answer 11

Yes.. just a second

Answer 12

Ok, so is it x=1 and x=2/3?

Answer 13

ok, good. Now, we know the denominator goes to zero as x goes to 1 and as x goes to 2/3, where does the numerator go? Does it go to zero as well?
(Evaluate the numerator at x = 1 and at x = 2/3)

Answer 14

So instead of making the numerator = to 0. I replace the x's with 1, and then 2/3 and see if that = to zero?

Answer 15

Yes. If they happen to be equal to 0 at 1 or at 2/3, then you may not have an asymptote at either. If they are nonzero, then you have a vertical asymptote at x = 1 and x = 2/3

Answer 16

No neither one will make it = to zero.. is that right?

Answer 17

Yes.

Answer 18

Ok :) What's next?

Answer 19

Oh or was that it?

Answer 20

Well, it means you have vertical asymptotes at x=1 and x=2/3
Now find the horizontal asymptotes

Answer 21

Well since there isnt a horizontal is it just y=0?

Answer 22

What do you mean by "there isn't a horizontal" ?

Answer 23

The horizontal asymptote.. since the numerator couldn't = to zero?

Answer 24

If the numerator is zero, then you've simply found a root of the function
(A root is a point where your function intersects the x axis)

Answer 25

Equating the numerator to 0 is not going to get you your horizontal asymptote :)

Answer 26

Oh okay :) So how do I go about finding the horizontal asymptote them?