If _ x x ( x^2 + ax - 2 x^2 + 4x + x^2 + cx ) = -1 , where a and c are real numbers such that a < c , then the value of c - a is equal to

Step-by-step solution

Given limit is _ x x ( x^2 + ax - 2 x^2 + 4x + x^2 + cx ) = -1 . Factor out x from each square root: _ x x^2 ( (1 + a x )^ 1/2 - 2 (1 + 4 x )^ 1/2 + (1 + c x )^ 1/2 ) = -1 Using the binomial expansion (1 + y)^ 1/2 = 1 + y 2 - y^2 8 + for small y (where y = k x ): (1 + a x )^ 1/2 = 1 + a 2x - a^2 8x^2 + -2 (1 + 4 x )^ 1/2 = -2 (1 + 2 x - 16 8x^2 ) = -2 - 4 x + 4 x^2 + (1 + c x )^ 1/2 = 1 + c 2x - c^2 8x^2 + Summing these expansions: 1 - 2 + 1 + 1 x ( a 2 - 4 + c 2 ) + 1 x^2 (- a^2 8 + 4 - c^2 8 ) Substitute this bac