How is this simplified?

Question

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

Multiply both numerator and denominator by the conjugate of numerator. — Claude Leibovici, Jan 26 '14 at 15:55
By the way, this is how you must write it when programing if you want to avoid significant losses of accuracy. — Claude Leibovici, Jan 26 '14 at 15:57
Se http://math.stackexchange.com/questions/652245/convergence-proof-lim-x-rightarrow-infty-sqrt4xx2-sqrtx2x/652316#652316 for a generalization. — Martín-Blas Pérez Pinilla, Jan 26 '14 at 23:39

Cure · Answer 1 · 2014-01-26T23:30:28.503

3

Hint: Multiply and divide by $\sqrt{x+h}+\sqrt{x}$

edited Jan 26 '14 at 23:30

answered Jan 26 '14 at 15:55

Cure

4,051

score 1 · Answer 2 · answered Jan 26 '14 at 15:58

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

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

How is this simplified?

2 Answers2