E is a true statement. If the function is continuous as x becomes very large, it may or may not have a horizontal asymptote at y = 2, but since nothing is known about the function other than its limit as x → ∞, you cannot say with certainty that the function is asymptotic to y = 2. Consider the function f(x) = 2. The limit as x approaches infinity is 2, but the function does not have a horizontal asymptote at y = 2.
B is a true statement. Since x = ∞ is not in the domain of the function, the only way that lim(x→∞) f(x) can exist is if the function is continuous as x approaches ∞.
The other statements may of may not be true, but lim(x→∞) f(x) = 2 does not require that any of them MUST be true.