Explain why f[x] = (3 x - 1)/Sqrt[x^2 + x +1] has two different horizontal asymptotes.

Answer:

The trick here is understanding how the dominant terms actually work when x is large and negative. Let's do the easy one first:

If x is large and positive, then Sqrt[x^2 + x +1] is close to Sqrt[x^2] = x, and the dominant term behavior of f[x] is like that of
(3 x)/Sqrt[x^2] = (3 x)/x = 3

Hence, we write f[x] =3 and have y = 3 as an asymptote.

If x is large and negative, then Sqrt[x^2 + x +1] is close to Sqrt[x^2] = -x because -x is the positive square root of x^2. Hence, the dominant term behavior of f[x] is like that of
(3 x)/Sqrt[x^2] = (3 x)/(-x)= 3/(-1) = - 3

Hence, we write f[x] = -3 and have y = - 3 as an asymptote.