A proposition with functions

Task number: 2784

Does the function \(f(x)=\sin x\) satisfy the following proposition, or its negation? \[ \forall \varepsilon>0\ \exists K>0\ \forall x: x>K \Rightarrow |f(x)|<\varepsilon.\]

  • Resolution

    The negation of the proposition reads: \( \exists \varepsilon>0\ \forall K>0\ \exists x: x>K \land |f(x)|\ge \varepsilon\).

    This negated proposition is satisfied by choosing \(\varepsilon=\frac12\) and \(x=\frac\pi2+\lceil K\rceil\pi\).

  • Result

    The proposition is not true for the given function.

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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