Evaluate $\lim_{x\to 0} {x\sqrt{y^2-(y-x)^2}\over (\sqrt{8xy-4x^2}-\sqrt{8xy})^3}$ Could any one tell me how to solve this limit?I tried doing rationalizations and all others, but failed. Thank you
$$\lim_{x\to 0} {x\sqrt{y^2-(y-x)^2}\over (\sqrt{8xy-4x^2}-\sqrt{8xy})^3}$$

If $y\ne0$ then both numerator and denominator have a factor $x\sqrt{x}$.  After cancelling that, the limit of the numerator is $\sqrt{2y}$ and the limit of the denominator is $0$, so your question has no limit.
If $y=0$ then the expression is $ix^2/(2ix)^3$ which has no limit either.
