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Math Help - Need help with eliminating indeterminate form.

  1. #1
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    [Limits] Need help with eliminating indeterminate form.

    Hello again.

    I need to eliminate some uncertainties. I know decent amount of trigonometric/algebraic but the reason I cant see what and how to use is probably that I haven't seen example usages of these enough. Here are some limit's I' would like to lean to calculate or to eliminate indeterminate form to be more exact.

    (L'hospital not allowed)
    1)\lim\limits_{x\rightarrow0} \dfrac{\tan^22x\cdot\ln(1-3x)}{\cos3x\cdot\arcsin^3x}
    I tried cos(3x)= cos^3(x)- 3sin^2(x)cos(x) and tan(2x)= 2tan(x)/(1- tan^2(x)). But they didnt help. Need some other ideas.

    (L'hospital not allowed)
    2)\lim\limits_{x\rightarrow0} \left(\dfrac{x-3}{x+2}\right)^{3x-5}
    No idea how to solve this to be honest

    (L'hospital not allowed)
    3)\lim\limits_{x\rightarrow0} \dfrac{1-\cos3x}{x\tan2x}
    I tried cos(3x)= cos^3(x)- 3sin^2(x)cos(x) and tan(2x)= 2tan(x)/(1- tan^2(x)). But they didnt help. Need some other ideas.
    Last edited by rain1; July 26th 2013 at 07:45 AM.
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  2. #2
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    Re: Need help with eliminating indeterminate form.

    For (1), you can break up the limit into parts:

    \lim_{x \to 0} \frac{\tan^2{2x}\ln(1-3x)}{\cos{3x}\sin^{-3}x} =

    \lim_{x \to 0} -12 \left(\frac{\tan{2x}}{2x}\right)^2 \frac{\ln(1-3x)}{-3x} \frac{1}{\cos{3x}} \left(\frac{x}{\sin^{-1}{x}}\right)^3 = -12.

    For (2), you can just plug in x=0 giving \left(\frac{-3}{2}\right)^{-5} = \left(-\frac{2}{3}\right)^5 = -\frac{32}{243}.

    For (3), you need to use the identity \sin^2{u}=\frac{1-\cos{2u}}{2} to get

    \lim_{x\to 0} \frac{1-\cos{3x}}{x\tan{2x}} =

    \lim_{x\to 0} \frac{2\sin^2{\frac{3}{2}x}}{x\tan{2x}}

    and then break it up like (1) to get the answer \frac{9}{4}.

    - Hollywood
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  3. #3
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    Re: Need help with eliminating indeterminate form.

    Thanks.

    \lim_{x \to 0} -12 \left(\frac{\tan{2x}}{2x}\right)^2 \frac{\ln(1-3x)}{-3x} \frac{1}{\cos{3x}} \left(\frac{x}{\sin^{-1}{x}}\right)^3 = -12.
    Where did -12 come from?

    That's nice that you found that trigonometric formula for 3. I sill didn't understand how exactly did you get 2 sin^2 3/2 x.
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  4. #4
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    Re: Need help with eliminating indeterminate form.

    Quote Originally Posted by rain1 View Post
    Thanks.

    \lim_{x \to 0} -12 \left(\frac{\tan{2x}}{2x}\right)^2 \frac{\ln(1-3x)}{-3x} \frac{1}{\cos{3x}} \left(\frac{x}{\sin^{-1}{x}}\right)^3 = -12.
    Where did -12 come from?

    That's nice that you found that trigonometric formula for 3. I sill didn't understand how exactly did you get 2 sin^2 3/2 x.
    We know by trigonometry that:

    \cos(2x) = 1 - 2\sin^{2}(x)\,\, \therefore\,\,\cos(3x) = \cos\left(2*\frac{3x}{2}\right) = 1 - 2\sin^{2}\left(\frac{3x}{2}\right)

    \text{ Thus:   } 1 - \cos\left(2*\frac{3x}{2}\right) = 2\sin^{2}\left(\frac{3x}{2}\right)

    Hope it helps.
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  5. #5
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    Re: Need help with eliminating indeterminate form.

    On #1, when I split up the expression, I did it so that each fraction has a limit of 1. So in the first fraction, I have 2x in the denominator and in the second fraction I have -3x in the denominator.

    The x's cancel out: x^2 in the denominator, x in the denominator, nothing, and x^3 in the numerator.

    I had to multiply by -12 to get the constants to cancel out: they were: 2^2 in the denominator, -3 in the denominator, nothing, and nothing. So -12 = 2^2 * (-3).

    Hope that helps.

    - Hollywood
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