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Limit with tangent

  1. Jan 25, 2008 #1
    1. The problem statement, all variables and given/known data

    Solve the following limit:

    [tex]\lim_{x \rightarrow 1}(1-x)\tan{\left(\frac{\pi x}{2}\right)}[/tex]

    3. The attempt at a solution

    I solved it using L'Hospital rule, it's equal to 2/pi, but is there any other way how to solve it? thanks a lot!

    The same question would apply to

    [tex]\lim_{x \rightarrow \frac{\pi}{3}}\left(\frac{\sin{\left(x-\frac{\pi}{3}\right)}}{1-2\cos{x}}\right)[/tex]
     
    Last edited: Jan 25, 2008
  2. jcsd
  3. Jan 27, 2008 #2
    Well, I've finally found a different way how to solve it... I post it here since it can be useful to someone..

    I substituted x with y = x-1, and then after some steps I've arrived at the following limit:

    [tex]\lim_{y \rightarrow 0}\frac{\frac{\pi}{2}y}{\sin{\frac{\pi}{2}y}}\frac{2}{\pi}\cos{\frac{\pi}{2}y} = \frac{2}{\pi}[/tex]
     
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