If the function f(x) = 2 2 - ( x + x)^3 1 - 2x is continuous at x = 4 , then the value of f ( 4 ) is ...

Step-by-step solution

Since f(x) is continuous at x = 4 , we have f ( 4 ) = _ x /4 f(x) . Let t = x + x . As x 4 , t ( 4 ) + ( 4 ) = 2 . Squaring both sides, t^2 = ( x + x)^2 = ^2 x + ^2 x + 2 x x = 1 + 2x . This gives 2x = t^2 - 1 . Substituting these into the limit: _ x /4 2 2 - ( x + x)^3 1 - 2x = _ t 2 2 2 - t^3 1 - (t^2 - 1) = _ t 2 2 2 - t^3 2 - t^2 This is a 0 0 form. Applying L'Hopital's rule: _ t 2 -3t^2 -2t = _ t 2 3t 2 = 3 2 2 Alternatively, by factorizing: _ t 2 ( 2 - t)(2 + 2 t + t^2) ( 2 - t)( 2 + t) = _ t 2 2 + 2 t + t^2