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Math Help - Algebra involved in limit...have the work, but need explanation

  1. #1
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    Exclamation Algebra involved in limit...have the work, but need explanation

    http://i53.tinypic.com/vgr8yp.jpg

    ^see pic

    Thank you so much!!
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  2. #2
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    They've multiplied by a cleverly disguised 1.

    In this case, \displaystyle\frac{\frac{1}{x}}{\frac{1}{x}}.


    So \displaystyle{\frac{x}{\sqrt{x^2 + x} + x} = \frac{x}{\sqrt{x^2+x}+x}\cdot \frac{\frac{1}{x}}{\frac{1}{x}}}

    \displaystyle{ = \frac{1}{\frac{\sqrt{x^2+x}+x}{x}}}

    \displaystyle{ = \frac{1}{\frac{\sqrt{x^2+x}}{x} + 1}}

    \displaystyle{ = \frac{1}{\frac{\sqrt{x^2+x}}{\sqrt{x^2}} + 1}}

     \displaystyle{= \frac{1}{\sqrt{\frac{x^2 + x}{x^2}} + 1}}

     \displaystyle{= \frac{1}{\sqrt{1 + \frac{1}{x}} + 1}}.
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  3. #3
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    Ahh I see, thanks!! I wish I had been that perceptive and noticed that trick :[
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