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$$\lim_{x\to 0} {a-\sqrt{a^2-x^2}\over x^2} =$$

$${a-\sqrt{a^2-x^2}\over x^2}\cdot{a+\sqrt{a^2-x^2}\over a+\sqrt{a^2-x^2}} = $$

$$a^{2} - a^{2} - x^{2}\over ax^{2} + x^{2}\sqrt{a^{2}-x^{2}} $$

$$-x^{2}\over ax^{2}+x^{2}\sqrt{a^{2}-x^{2}}$$

$$-1\over a+\sqrt{a^{2}-x^{2}}$$

Can anyone tell me where I'm wrong?

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  • $\begingroup$ $a^2 - (a^2-x^2) = x^2$ $\endgroup$ Oct 5, 2014 at 23:55
  • $\begingroup$ Check your algebra. $\endgroup$ Oct 5, 2014 at 23:57

2 Answers 2

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Should be $${a^{2} - (a^{2} - x^{2})\over ax^{2} + x^{2}\sqrt{a^{2}-x^{2}}}= {a^{2} - a^{2} +x^{2}\over ax^{2} + x^{2}\sqrt{a^{2}-x^{2}}}.$$

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Third equality $$\left (a - \sqrt{a^2-x^2}\right )\left (a+\sqrt{a^2+x^2}\right ) \color{blue}{=x^2}$$

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