When x is so small that its square and its higher powers may be neglected, then the value of (1+ 3 4 x )⁻⁴ (3+x) (3-x)^3 is approximately equal to

Step-by-step solution

Given that x^n 0, n=2,3,4 Let y= (1+ 3 4 x )⁻⁴ (3+x) (3-x)^3 aligned & = (1+ 3 4 x )⁻⁴ 3 (1+ 1 3 x )^ 1 / 2 (3)^ 3 / 2 (1- 1 3 x )^ 3 / 2 & y=3^ 1 / 2-3 / 2 (1+ 3 4 x )⁻⁴(1+3 x)^ 1 / 2 (1- 1 3 x )^ -3 / 2 & y= 1 3 [1+(-4) ( 3 4 x )+ (-4)(-5) 2 ! ( 3 4 x )^2+ ] & [1+ (- 3 2 ) ( -1 3 x )+ (-3 / 2)(-3 / 2-1) 2 ! (- 1 3 x )^2+ ] aligned Now we will neglect term with x^2 and greater powers. i.e. x^2=x^3=x^4= =x^n 0 . aligned & y= 1 3 [1-3 x+0+ ] [1+ 1 6 x-0+ ] & y= 1 3 (1-3 x) (1+ 1 6 x ) (1+ 1 2 x ) & y= 1 3 (1+ 1 6 x-