Question

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

• A. $\frac{1}{128}$
• B. $\frac{1}{64}$
• C. $\frac{1}{2\sqrt{2}}$
• D. none of these

Answer 1

The correct option is A $\frac{1}{128}$

$\underset{x\to 0}{lim}\frac{xy\sqrt{y^2-(y-x{)}^2}}{(\sqrt{8xy-4x^2}+\sqrt{8xy}{)}^3}=\underset{x\to 0}{lim}\frac{xy\sqrt{2xy-x^2}}{(\sqrt{8xy-4x^2}+\sqrt{8xy}{)}^3}=\underset{x\to 0}{lim}\frac{y\sqrt{2y-x}}{(\sqrt{8y-4x}+\sqrt{8y}{)}^3}=\frac{\sqrt{2}{y}^{3/2}}{(2\sqrt{8y}{)}^3}=\frac{\sqrt{2}}{8\times 8^{3/2}}=\frac{1}{128}$
Hence, option 'A' is correct.