Question

Solve:
${lim}_{x\to 0}\frac{x\sqrt{y^2-(y-x{)}^2}}{{\{\sqrt{(8xy-4x^2)}+\sqrt{8xy}\}}^3}$

Answer 1

${lim}_{x\to 0}\frac{x\sqrt{y^2-{(y-x)}^2}}{{(\sqrt{8xy-4x^2}+\sqrt{8xy})}^3}$

$={lim}_{x\to 0}\frac{x\sqrt{(y-y+x)(y+y-x)}}{{(2\sqrt{x(2y-x)}+2\sqrt{2y}\sqrt{x})}^3}$

$={lim}_{x\to 0}\frac{x\sqrt{x(2y-x)}}{8x\sqrt{x}{(\sqrt{2y-x}+\sqrt{2y})}^3}$

$={lim}_{x\to 0}\frac{\sqrt{(2y-x)}}{8{(\sqrt{2y-x}+\sqrt{2y})}^3}$

$=\frac{\sqrt{(2y-0)}}{8{(\sqrt{2y-0}+\sqrt{2y})}^3}$

$=\frac{\sqrt{2y}}{8{\left(2\sqrt{2y}\right)}^3}$

$=\frac{\sqrt{2y}}{8\times 8\times 2y\sqrt{2y}}$

$=\frac{1}{128y}$