But How Many Digits Are There?

Calculus Level 3

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

Let \(x_n = 1.\underbrace{000000000\ldots 0}_{n \text{ number of 0's}}1 \) and \(y_n = 0.\underbrace{999999999\ldots9}_{(n+1) \text{ number of 9's}} \).

Compute the limit above.

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