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I am studying up on my calculus and I have found this limit and I am not sure how to approach this without using L'Hopital's Rule. I am guessing there is an issue with my algebra. Here it is. $$\lim_{x\to 0}\frac{\sqrt{x-x\sqrt{x+x^2}-x^2}}{\sqrt x}$$I am lost on this one! Please help me on this. Thanks.

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closed as off-topic by Ahaan S. Rungta, user7530, user98602, ronno, Ali Caglayan Dec 28 '14 at 0:50

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  • $\begingroup$ Show us what you have tried please. $\endgroup$ – Ahaan S. Rungta Dec 27 '14 at 22:14
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    $\begingroup$ $\lim_{x\to 0} \sqrt{1-\sqrt{x+ x^2}-x}$ $\endgroup$ – MBYagbasan Dec 27 '14 at 22:17
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Note that $$\lim_{x\to 0}\frac{\sqrt{x-x\sqrt{x+x^2}-x^2}}{\sqrt x}=\lim_{x\to 0}\sqrt{\frac{x-x\sqrt{x+x^2}-x^2}{x}}$$Then we have $$=\lim_{x\to 0}\sqrt{1-\sqrt{x+x^2}-x}=1$$

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  • $\begingroup$ Duh! I feel like an idiot. Anyways, thanks alot. $\endgroup$ – some_user Dec 27 '14 at 22:30
  • $\begingroup$ No problem. Sometimes its the simple things that we get stuck on :) $\endgroup$ – homegrown Dec 27 '14 at 22:31

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