Variance of predicted value in a linear regression when $n \to \infty$ The following question is from Kutner's Applied Linear Statistical Models - Ch 2 - 2.12

To answer the question a few pieces of information are needed, provided below:







What I gather the question is asking me is that if I take the limit as $n \to \infty$ then what happens to the variance of my new prediction?
Now clearly based on the expressions below, it would appear that in both cases $Var(pred)$ and $Var(\hat{Y_{h}})$ would both be $0$. And as such they can be brought increasingly close to $0$.
But the solution says that this is not the case and I'm wondering why? Is it because they specified the theoretical variances? i.e $\sigma^{2}(pred)$ and $\sigma^{2}(\hat{Y_{h}})$ and I took the estimates by using the sample variances for the respective pieces? But even still the theoretical variances still involve having an $n$ in the denominator.....Am I looking at something wrong?
 A: As a preliminary note, I'm not a fan of the notation used in the excerpted material, so I'll avoid using most of it.  In any case, as you can see from the formulae, the difference between the variance of the prediced value and the variance of a new value is the variance of the error term in the regression, which is estimated by the $\text{MSE}$ term.  So the natural mathematical question here is, what is the limit $\lim_{n \rightarrow \infty} \text{MSE}_n$?  (Hint: it is not generally equal to zero.)  If you can answer that then you can find the difference in the limits of the two variances.
Stepping back from the mathematics of this, to look at the intuition, it is worth noting that a new value in the regression model will have variability that depends on two things.  There is some variability coming from the fact that you don't know the true expected value in the regression (and you estimate this with the predicted value) and then there is additional variance coming from the fact that the new observation will deviate from the true expected value because of the "error term".  (This occurs by definition; we define the error term in the model as the deviation of the value from its true expectation.)
If you have a careful look at the behaviour of the MSE, you should be able to find the two variance limits of interest, and it should be possible to explain this difference intuitively by considering the difference between a new point in the regression and the expected value of the response variable for that point.
