Find the asymptotes.

Question

r(x)= 4x²+4 / x²+6x+9

vertical asymptotes=

horizontal asymptotes=

Answer 1

ok so then we solve BOTH problems...

As written the way Edward stated, the vertical asymptote is at x=0, since that is the only denominator in the problem. Otherwise the x^2 term dominates and the function is quadratic, and as such, goes off to infinity

We SUSPECT the function is f(x) = (4x^2 +4)/(x^2 + 6x + 9) <--- notice the parenthesis

which factors as = 4(x^2+1)/(x+3)^2

In this case, the vertical asymptote is at x=-3.

4

______________________

Dividing these polynomials x^2 + 6x + 9 | 4x^2 + 0x + 4

4x^2 + 24x + 36

------------------------------

-24x - 32

So the function can be rewritten as: f(x) = 4 + [ -8(3x+4)/(x+3)^2 ]

the quadratic in the denominator of the remainder dominates, so the the remainder fraction goes to zero

as x goes off to infinity. therefore the entire remainder vanishes as x approaches infinity, so the horizontal asymptote is 4