IIT JEE , AIEEE Forum - Mathematics , Differential Calculus - Q> Ltx-0 (x?{y^2-(y-x)^2 })/([?(8xy-4x^2)+?8xy])^3

iitkgp_bipin

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

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

Taking x^3/2 common from numerator and denominator and cancelling them, we get :

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

Putting x=0 in the above expression we get

Limit = 1/8y

allamraju

rohit_08_vns

The answer is 1/4.