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I have the following problem:

$$\frac{\sqrt{x+h}-\sqrt x}{h}$$

Wolfram reduces it to:

$$\frac{1}{\sqrt{h+x}+\sqrt x}$$

http://www.wolframalpha.com/input/?i=%28%28x%2Bh%29^%281%2F2%29+-+x^%281%2F2%29%29+%2F+h

But I can't think of how it comes to this conclusion. I'm really curious as to how this is solved. Any hints would be very helpful.

Thanks

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  • $\begingroup$ Multiply both numerator and denominator by the conjugate of numerator. $\endgroup$ – Claude Leibovici Jan 26 '14 at 15:55
  • $\begingroup$ Hint: use identity $a^2-b^2=(a-b)(a+b)$. $\endgroup$ – alans Jan 26 '14 at 15:56
  • $\begingroup$ By the way, this is how you must write it when programing if you want to avoid significant losses of accuracy. $\endgroup$ – Claude Leibovici Jan 26 '14 at 15:57
  • $\begingroup$ Se math.stackexchange.com/questions/652245/… for a generalization. $\endgroup$ – Martín-Blas Pérez Pinilla Jan 26 '14 at 23:39
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Hint: Multiply and divide by $\sqrt{x+h}+\sqrt{x}$

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Multiply both the numerator and denominator by the conjugate $\sqrt{x+h} + \sqrt x$:

$$\frac{\sqrt{x+h} - \sqrt x}{h}\cdot\frac{\sqrt{x+h} + \sqrt x}{\sqrt{x+h} + \sqrt x}\\ \frac{x+h+\sqrt{x+h}\sqrt x-\sqrt{x+h}\sqrt x-x}{h(\sqrt{x+h}+\sqrt x)}\\ \frac{h}{h(\sqrt{x+h}+\sqrt x)}=...$$

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