But How Many Digits Are There?

Calculus Level 3

limn10n+1(xnxnynyn)\large \displaystyle \lim_{n\to\infty}10^{n+1}\left(x_n^{x_n} - y_n^{y_n}\right)

Let xn=1.0000000000n number of 0’s1x_n = 1.\underbrace{000000000\ldots 0}_{n \text{ number of 0's}}1 and yn=0.9999999999(n+1) number of 9’sy_n = 0.\underbrace{999999999\ldots9}_{(n+1) \text{ number of 9's}} .

Compute the limit above.

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