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Math Help - finding limits as x tends to pi/4

  1. #1
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    finding limits as x tends to pi/4

    how do you find the lim of ( cosec^2 x -2)/ ( cot x -1 ) as x tends to pi/4?

    thanks!
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  2. #2
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    Quote Originally Posted by alexandrabel90 View Post
    how do you find the lim of ( cosec^2 x -2)/ ( cot x -1 ) as x tends to pi/4?

    thanks!
    Remember that \sin^2{x} + \cos^2{x} = 1

    Dividing both sides by \sin^2{x} gives us

    1 + \cot^2{x} = \csc^2{x}.


    So \frac{\csc^2{x} - 2}{\cot{x} - 1} = \frac{1 + \cot^2{x} - 2}{\cot{x} - 1}

     = \frac{\cot^2{x} - 1}{\cot{x} - 1}

     = \frac{(\cot{x} - 1)(\cot{x} + 1)}{\cot{x} - 1}

     = \cot{x} + 1.


    So \lim_{x \to \frac{\pi}{4}}\frac{\csc^2{x} - 2}{\cot{x} - 1} = \lim_{x \to \frac{\pi}{4}}(\cot{x} + 1)

     = \cot{\frac{\pi}{4}} + 1

     = 1 + 1

     = 2.
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