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Help With Trig Inequality

  • Thread starter henry22
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  • #1
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Homework Statement


I am attempting to show that [itex]-x \leq sin(x) \leq x[/itex] for x>0 and thus [itex]\int^1_0 nxsin(\frac{1}{nx})dx[/itex] converges to 1.


Homework Equations



I know that I need to use the fact that I have shown that the limit as T tends to infinity of [itex]\int^T_1 \frac{cos(x)}{\sqrt{x}}dx[/itex] exists.


The Attempt at a Solution


Homework Statement





Homework Equations





The Attempt at a Solution

 

Answers and Replies

  • #2
LCKurtz
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Homework Statement


I am attempting to show that [itex]-x \leq sin(x) \leq x[/itex] for x>0 and ...
The -x part is trivial and for the rest integrate both sides of cos(t) ≤ 1 between 0 and x, x > 0.
 
  • #3
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The -x part is trivial and for the rest integrate both sides of cos(t) ≤ 1 between 0 and x, x > 0.
OK I've done this and I get the inequality I need. But can I just check, I don't understand how I have used the equation I need to in the OP?

For the second part if I know that -x<= sinx <= x then -1<=nx(sin(1/nx)) <= 1 but then I'm a bit stuck
 

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