limit as h goes to 0 of$\displaystyle \frac{f(2+h)-f(2)}{h}$ where f(x) = $\displaystyle \sqrt{x}$
It looks similar to differentiation from first principles, but I can't see how to use that...
$\displaystyle \lim_{h \to 0} \frac{\sqrt{2 + h} - \sqrt{2}}{h}$
Rationalize the numerator:
$\displaystyle = \lim_{h \to 0} \frac{\sqrt{2 + h} - \sqrt{2}}{h} \cdot \frac{\sqrt{2 + h} + \sqrt{2}}{\sqrt{2 + h} + \sqrt{2}}$
$\displaystyle = \lim_{h \to 0} \frac{2 + h - 2}{h(\sqrt{2 + h} + \sqrt{2})}$
$\displaystyle = \lim_{h \to 0} \frac{h}{h(\sqrt{2 + h} + \sqrt{2})}$
$\displaystyle = \lim_{h \to 0} \frac{1}{\sqrt{2 + h} + \sqrt{2}}$
Now you can evaluate the limit by direct substitution.
01
EDIT: too slow...