To the Limit, If It Exists
I saw, as one might see the transit of Venus, a quantity passing through infinity and changing its sign from plus to minus. I saw exactly how it happened … but it was after dinner and I let it go.—SIR WINSTON CHURCHILL, My Early Life (1930)
At first thought, the peculiar behavior of the expression (1 + 1/n)n for large values of n must seem puzzling indeed. Suppose we consider only the expression inside the parentheses, 1 + 1/n. As n increases, 1/n gets closer and closer to 0 and so 1 + 1/n gets closer and closer to 1, although it will always be greater than 1. Thus we might be tempted to conclude that for “really large” n (whatever “really large” means), 1 + 1/n, to every purpose and extent, may be replaced by 1. ...