19) We know that $x=0.9 \ldots$ and that $1 / x=1.1 \ldots$ (the dots indicate that the further decimal digits have been omitted). What is the digit that comes immediately after the 9 in the decimal expansion of $x$?

$\begin{array}{llll}\text { (A) } 0 & \text { (B) } 1 & \text { (C) } 9 & \text { (D) cannot be determined uniquely }\end{array}$ (E) none of the above.
19) The answer is (A).

Since $\frac{1}{x}=1.1 \ldots \geq \frac{11}{10}$, we have $0.9 \leq x \leq \frac{10}{11}=0.909 \ldots$ and thus the second decimal digit is 0.