Using Monotonicity to Find the Range of a Function
For each posi...
Question
For each positive real number λ, let Aλ be the set of all natural numbers n such that |sin(√n−1)–sin(√n)|<λ. Let Acλ be the complement of Aλ in the set of all natural numbers. Then
A
A12,A13,A25 are all finite sets
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B
A12 is a finite set but A13,A25 are infinite sets
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C
Ac12,Ac13,Ac25 are all finite sets
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D
A13,A25 are finite set but A12 is an infinite set
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Solution
The correct option is CAc12,Ac13,Ac25 are all finite sets As n→∞, |sin(√n−1)–sin(√n)|→0 Therefore, there can infinite number of values of λ such that |sin(√n−1)–sin(√n)|<λ, where λ>0 ∴A12,A13,A25 are all infinite sets ⇒Ac12,Ac13,Ac25 are all finite sets