# Reelle Zahlen/Nullfolgen/Unterschiedliches Verhalten der inversen Folge/Aufgabe/en

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Give examples of convergent sequences of real numbers ${\displaystyle {}{\left(x_{n}\right)}_{n\in \mathbb {N} }}$ and ${\displaystyle {}{\left(y_{n}\right)}_{n\in \mathbb {N} }}$ with ${\displaystyle {}x_{n}\neq 0}$, ${\displaystyle {}n\in \mathbb {N} }$, and with ${\displaystyle {}\lim _{n\rightarrow \infty }x_{n}=0}$ such that the sequence

${\displaystyle \left({\frac {y_{n}}{x_{n}}}\right)_{n\in \mathbb {N} }}$
1. converges to ${\displaystyle {}0}$,
2. converges to ${\displaystyle {}1}$,
3. diverges.