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Math Help - Evaluate limits involving indeterminate forms

  1. #1
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    Evaluate limits involving indeterminate forms

    Evaluate the limit:

    lim t->0 (t^9)/(tan 2t)^9

    Do I have to keep applying l'hopital's rule 9 times or is there another way to do it?
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  2. #2
    ynj
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    no, you may use Taylor.
    \lim\frac{t^9}{\tan^9 2t}=\lim\frac{t^9}{(2t+o(t^2))^9}=\lim\frac{t^9}{5  12t^9+o(t^9)}=lim\frac{1}{512+\frac{o(t^9)}{t^9}}=  \frac{1}{512}
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  3. #3
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    Helolo, dalbir4444!

    Use the theorem: . \lim_{\theta\to0}\frac{\sin\theta}{\theta} \:=\:1 \qquad \lim_{\theta\to0}\frac{\theta}{\sin\theta} \:=\:1


    Evaluate: . \lim_{t\to0}\frac{t^9}{(\tan 2t)^9}
    \text{We have: }\;\frac{t^9}{\dfrac{(\sin2t)^9}{(\cos2t)^9}} \;=\; \frac{t^9(\cos2t)^9}{(\sin2t)^9}

    Multiply by \frac{2^9}{2^9}\!:\quad \frac{2^9}{2^9}\cdot \frac{t^9(\cos2t)^0}{(\sin2t)^9} \;=\;\frac{(\cos2t)^9}{2^9}\cdot\frac{2^9t^9}{(\si  n2t)^9} \;=\;\left(\frac{\cos2t}{2}\right)^9 \cdot\frac{(2t)^9}{(\sin2t)^9} . = \;\left(\frac{\cos2t}{2}\right)^9\left(\frac{2t}{\  sin2t}\right)^9


    \text{Take the limit: }\;\lim_{t\to0}\left(\frac{\cos2t}{2}\right)^9\lef  t(\frac{2t}{\sin2t}\right)^9 \;=\; \lim_{t\to0}\underbrace{\left(\frac{\cos2t}{2}\rig  ht)}_{\text{this is }\frac{1}{2}}\,\!\!^9\cdot\lim_{t\to0}\underbrace{  \left(\frac{2t}{\sin2t}\right)}_{\text{This is 1}}\,\!\!^9

    . . . . . . . . . . = \;\left(\frac{1}{2}\right)^9\cdot1^9 \;=\;\frac{1}{512}

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