Evaluating the limit $\lim_{h\to 0} \frac{\sqrt {1 - (x+h) ^2} - \sqrt {1-x^2}} {h}$ Find the limit
$\lim_{h\to 0} \frac{\sqrt {1 - (x+h) ^2} - \sqrt {1-x^2}} {h}$
I tried substituting "h" and also multiplying $\frac{\sqrt {1 - (x+h) ^2} + \sqrt {1-x^2}}{\sqrt {1 - (x+h) ^2} + \sqrt {1-x^2}}$

Your idea is fine, indeed we obtain
$$\frac{\sqrt {1 - (x+h) ^2} - \sqrt {1-x^2}} {h}\cdot \frac{\sqrt {1 - (x+h) ^2} + \sqrt {1-x^2}}{\sqrt {1 - (x+h) ^2} + \sqrt {1-x^2}}=$$
$$=\frac{{1 - (x+h) ^2} -  ({1-x^2})}{h(\sqrt {1 - (x+h) ^2} + \sqrt {1-x^2})}=$$
$$=\frac{-2xh-h^2}{h(\sqrt {1 - (x+h) ^2} + \sqrt {1-x^2})}$$
and since $h$ cancels out we can conclude.
