Horizontal Asymtotes are where Y cannot be.
If the top polynomial is greater, there is no horizontal asymtote.
If both polynomials are the same, divide the greatest top polynomial by the greatest bottom one.
If the bottom polynomial is greater, the horizontal asymtote is y = 0.

4. y = (6x - 9) / (x^2+4x+5)^(1/2)

We can take the bottom x^2 and square root it.

So the two leading coefficients would be 6x / (x^2)^(1/2) = 6.
Thereby as x approaches infinity we have 6(infinity) / ((infinity)^2)^(1/2) = 6.

5. y = 3x/ (x^4+1)^(1/4)

We find that the polynomials are basically the same as (x^4)^(1/4) = x
So we have an x on top and x on bottom.

3x / (x^4)^(1/4)

We solve and get both 3 and -3, because of the x^4 it can be any of the two.


6. y = 2x / (e^x+9)

If we evaluate as x approaches infinity, the bottom would be greater than the top as e^infinity would become a much larger infinity than 2(infinity)
This would make the limit around 0.

If we evaluate as x approaches -infinity, then e^x would get really small while the top would get much more negative.
This would make the limit around -infinity.