Question

$\lim_{h \0} \frac{ \sqrt{3+3h}- \sqrt{3} }{h}$

Answer 1

$\lim_{h\rightarrow0}\frac{\sqrt{3+3h}-\sqrt{3}}{h}=\\\\\lim_{h\rightarrow0}\frac{\sqrt{3+3h}-\sqrt{3}}{h}\times\frac{\sqrt{3+3h}+\sqrt{3}}{\sqrt{3+3h}+\sqrt{3}}=\\\\\lim_{h\rightarrow0}\frac{(3+3h)-3}{h(\sqrt{3+3h}+\sqrt{3})}=\\\\\lim_{h\rightarrow0}\frac{3h}{h(\sqrt{3+3h}+\sqrt{3})}=$

$\lim_{h\rightarrow0}\frac{3}{\sqrt{3+3h}+\sqrt{3}}=\\\\\frac{3}{\sqrt{3+3\cdot0}+\sqrt{3}}=\\\\\frac{3}{\sqrt{3}+\sqrt{3}}=\\\\\frac{3}{2\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\\\\\frac{3\sqrt{3}}{6}=\\\\\boxed{\frac{\sqrt{3}}{2}}$