Question

lim as h->0 of (sqrt(6(a+h)) -sqrt(6a))/h
What is the limit in terms of the constant a?

Answer 1

$\displaystyle \lim_{h \to 0} (\frac{\sqrt{6(a+h)}- \sqrt{6a}}{h}) = \lim_{h \to 0} (\frac{(\sqrt{6(a+h)}- \sqrt{6a})(\sqrt{6(a+h)}+ \sqrt{6a})}{h (\sqrt{6(a+h)}+ \sqrt{6a})}) =$

$\displaystyle= \lim_{h \to 0} (\frac{6(a+h)- 6a}{h (\sqrt{6(a+h)}+ \sqrt{6a})}) = \lim_{h \to 0} (\frac{6h}{h (\sqrt{6(a+h)}+ \sqrt{6a})}) =$

$= \displaystyle \lim_{h \to 0} (\frac{6}{ (\sqrt{6(a+h)}+ \sqrt{6a})}) = \frac{6}{2\sqrt{6a}} = \frac{\sqrt{6}}{2 \sqrt{a}} =\sqrt{ \frac{3}{2a}}$