Mathematics?

[minou]

The function:

sqrt(x^2+9x+16)-x

has one horizontal asymptote at y=?

[mathsmanretired]

It is y = 4.5.
This is because when x gets very large the 16 is tiny in comparison to the other terms so you can forget it.
sqrt(x^2 + 9x) = x*sqrt(1 + 9/x)
sqrt(1 + 9/x) = (1 + 9/x)^0.5
= 1 + 4.5/x using the binomial expansion. All higher terms can be ignored because they will have powers of x in the denominator even after the stage below.

So sqrt(x^2 + 9x) - x = x*(1 + 9/x)^0.5 - x
= x*(1 + 4.5/x) - x = 4.5