What is a bound of \frac{1}{\sin x} - \frac{1}{x} on

Papierskiix5n

Papierskiix5n

Answered question

2022-04-16

What is a bound of 1sinx1x on (0,a]?
Is 1sinx1x bounded in (0,a] , a<π2?

Answer & Explanation

superpms01wks1

superpms01wks1

Beginner2022-04-17Added 13 answers

For |t|π2 we have,
|1sint1t|12π
That's all we have for it. l 0tπ2
tantsintt2costsin2t+1t2=(1sint1t)0
as a result, the function, f:t1sint1t is increasing on (0,π2)
then for t(0,π2)
f(0)f(t)f(π2)
that is,
0=f(0)=limh0(1sinh1h)(1sint1t)f(π2)=12π
Due to the peculiar nature of the function, the conclusion is valid t(0,π2)

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