How to find limit of this function: $\lim_{n\to\infty} 0.99\ldots99^{10^n}$?

Question

How to find this limit:

$$ \lim_{n\to\infty} 0.99\ldots99^{10^n},$$

$$ \text{where number of 9 is } n. $$

My solution: $$ \text{if } n\to\infty \text{ then } 0.99\ldots99 \to\ 0.(9) $$ $$ \text{because } 0.(9) = 1 \implies \lim_{n\to\infty} 0.99\ldots99^{10^n}=1^{10^\infty} = 1 $$

But my solution is wrong. Why? How can I correct it?

Answer 1

The limit you're trying to find is

$$ \lim_{n\to\infty} (1-10^{-n})^{10^n} $$

Substitute $u \mapsto 10^n$ and you get

$$ \lim_{u\to\infty} \left(1-\frac 1 u \right)^u = \frac 1 e $$

Answer 2

Your answer is wrong because 1 to the power infinity is not equal to 1 but it has unspecified value just like 0/0