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Math Help - Limits of '0/0' functions

  1. #1
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    Limits of '0/0' functions

    How can find a limit, or prove it doesn't exist, for the following function:

    \lim_{x\rightarrow 0}{\frac{sin(x)}{\sqrt{1-cos(x)}}}

    When I differentiate, I get another function with a similar problem, and it just gets more complicated, so I can assume l'H˘pital is not what I need to use here.

    I tried to simplify the function by multiplying it by \frac{\sqrt{1+cos(x)}}{\sqrt{1+cos(x)}} , and then I got the function: \sqrt{1+cos(x)} but the changed the function and I guess I can't use it (moreover, it shows the limits are \sqrt{2} and -\sqrt{2}, and in the original function I found that it doesn't go straight to this point, but 'gets crazy' and reaches infinity when it gets closer to 0 (from both sides).
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  2. #2
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    Krizalid's Avatar
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    there's no limit because the one sided ones are different, since \underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{\sqrt{1-\cos x}}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+\cos x}\sin x}{\left| \sin x \right|}.
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  3. #3
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    Thank you very much !

    There's my mistake...
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