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Math Help - proving a limit by definition..

  1. #1
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    proving a limit by definition..

    \lim_{x\rightarrow\infty}\sqrt{2x-\sin3x}=\infty
    we need to prove that for every N>0 there is M>0 so for every x>M
    we have f(x)>N .
    i have a solution of a similar question:
    \lim_{x\rightarrow\infty}x^{2}+3=\infty
    the solution is:
    we take N\geq3 we find M that fits it.
    we clame that M= \sqrt{N-3}
    if x>M then x> \sqrt{N-3}
    so x^{2}+3>N
    i cant use the similar question because i cantisolate x on one side.
    and i cant use anything but thisdefinition to solve it
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  2. #2
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    but when X is big enought, 2X is big too, but sin(3X) remains between -1 and 1... doesn´t it lead you to somewhere?
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  3. #3
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    yes it leads to infinity but thats only word not a proof by definition like i showed
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  4. #4
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    Surely you can show that for x\ge 2,

    \sqrt {2x - 3}  < \sqrt {2x - \sin (3x)}  < \sqrt {2x + 3}~?
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  5. #5
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    Plato just showed what I meant ^^
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