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Limit of a trigonometric function (Advanced problem)

  1. Oct 22, 2009 #1
    Adminstrator: This is a double post, so please feel free to delete this one.

    The relevant post is "Limit of a Trigonometric Function (Involved Problem)".


    1. The problem statement, all variables and given/known data

    Evaluate [tex] \underset{x\to 0}{\mathop{\lim }}\,\frac{\sec \frac{x}{2}-1}{x\sin x} [/tex]

    2. Relevant equations



    3. The attempt at a solution

    Hello there,

    I tried to evaluate this limit using two different approaches, both of which still leave the limit in indeterminate form when 0 is substituted.

    Thank you for your help!

    I) Pure algebraic manipulation:

    [tex] \begin{align}
    & \underset{x\to 0}{\mathop{\lim }}\,\frac{\sec \tfrac{x}{2}-1}{x\sin x}=\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{1-\cos \tfrac{x}{2}}{\cos \tfrac{x}{2}}\div \frac{1}{x\sin x} \right)=\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{1-\cos \tfrac{x}{2}}{x\sin x\cos \tfrac{x}{2}} \right) \\
    & =\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{1-\cos \tfrac{x}{2}}{\tfrac{x}{2}} \right)\times \underset{x\to 0}{\mathop{\lim }}\,\left( \frac{\tfrac{1}{2}}{\sin x\cos \tfrac{x}{2}} \right) \\
    \end{align} [/tex]

    II) Manipulation with conjugate method:

    [tex] \begin{align}
    & \underset{x\to 0}{\mathop{\lim }}\,\frac{\sec \tfrac{x}{2}-1}{x\sin x}=\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{1-\cos \tfrac{x}{2}}{x\sin x\cos \tfrac{x}{2}} \right) \\
    & \text{Let u = }\tfrac{x}{2}.\text{ Since }\underset{u\to 0}{\mathop{\lim }}\,u=0\text{, u still tends to 0}\text{.} \\
    & \underset{u\to 0}{\mathop{\lim }}\,\frac{1-\cos u}{\tfrac{u}{2}\cos u\sin \tfrac{u}{2}}\left( \frac{1+\cos u}{1+\cos u} \right)=\underset{u\to 0}{\mathop{\lim }}\,\frac{1-{{\cos }^{2}}u}{\tfrac{u}{2}\cos u\sin \tfrac{u}{2}(1+\cos u)} \\
    & =\underset{u\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{2}}u}{\tfrac{u}{2}\cos u\sin \tfrac{u}{2}(1+\cos u)} \\
    \end{align} [/tex]
     
    Last edited: Oct 22, 2009
  2. jcsd
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