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Problem with limiits


A well known limit is:
Limit[(1 + 1/n)^n, n -> Infinity]=E
I tried it and it works... solution seems built in.

I tried:
Limit[(1 + 1/Prime[n])^Prime[n], n -> Infinity]

Again I tried:
Limit[(1 + 1/Prime[n])^Prime[n], n -> 2000]

Here's how I got an estimate:
Table[(1 + 1/Prime[n])^Prime[n], {n, 1, 400}];
ListPlot[%]

It appears to be approaching  E as well.
N[(1 + 1/Prime[2000])^Prime[2000], 100] - E
-0.000078156838841603507435562510935842641134579112458192281970712293762387821356624136556497567576200


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