1996 Antalya Matematik Olimpiyatı Soru 12

[matematikolimpiyati]

Aşağıdaki beş diziden kaç tanesinin limiti vardır?

$
\begin{array}{rcll}
\text{I.} & & 1,1,1,...,1,... \\
\text{II.} & & 0,1,0, \dfrac12,0, \dfrac13,...,0, \dfrac1n, 0, \dfrac{1}{n+1},... \\
\text{III.} & & (0,2),(0,22),(0,222),(0,2222),... \\
\text{IV.} & & \dfrac{\sin 1}{1}, \dfrac{\sin 2}{2}, \dfrac{\sin 3}{3},..., \dfrac{\sin n}{n},... \\
\text{V.} & & 0,\dfrac32, \dfrac{-2}{3}, \dfrac54, \dfrac{-4}{5},..., \left( (-1)^n+ \dfrac1n \right) ,...
\end{array}$

$\textbf{a)}\ 1 \qquad\textbf{b)}\ 2  \qquad\textbf{c)}\ 3 \qquad\textbf{d)}\ 4 \qquad\textbf{e)}\ 5$

[taftazani44]

ııı.
$\lim _{n\rightarrow \infty }\dfrac{22\ldots 2}{10^{n}}$
$=\lim _{n\rightarrow \infty }\dfrac{\dfrac{2}{9}\left( 10^{n}-1\right) }{10^{n}}$
$= \dfrac{2}{9}$
ıv.
$\begin{aligned}\lim _{n\rightarrow \infty }\dfrac{\sin }{n}\\ =\lim ^{-}\dfrac{1\leq \sin 1\leq 1}{n}\\ n\rightarrow \infty \\ =\dfrac{k}{\infty }=0\end{aligned}$
Diğerlerinin limitlerinin varlığı belli.