_ x 0 1+x x - x ^2 2 x =

Step-by-step solution

_ x 0 1+x x - x ^2 2 x ( 0 0 . form ) On rationalising aligned & = _ x 0 1+x x - x ^2 2 x 1+x x + x 1+x x + x & = _ x 0 1+x x- x ^2 2 x & _ x 0 1 1+x x + x & = _ x 0 (1- x)+x x ^2 2 x 1 2 & = _ x 0 2 ^2 x 2 +2 x x 2 x 2 ^2 2 x 1 2 & = _ x 0 2 ^2 x 2 [1+ x x 2 ] ^2 2 x 1 2 & = _ x 0 1 4 ( x 2 x 2 )^2 [1+ 2 x 2 x 2 ] 4 [ 2 x 2 x ]^2 & aligned On applying limits, we get = 1 4 (1)^2 [1+2 1] 4(1)^2 = 3 16 .