Question

`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4` is equal to

Options

• `z/(2^(11/3))`

• `1/(2^(23/3 . z))`

• `2^(21/3) z`

• None of these

Answer 1

`1/(2^(23/3 . z))`

Explanation:

`lim_(x -> 0) (xroot(3)(z^2 - (z - x)^2))/(root(3)(8xz - 4x^2) + root(3)(8xz))^4`

= `lim_(x -> 0) (xroot(3)(2xz - x^2))/(root(3)(x)root(3)(8z - 4x) + root(3)(8z) root(3)(x))^4`

= `lim_(x -> 0) (x^(4/3) root(3)(2z - z))/(x^(4/3) [root(3)(8z - 4x) + root(3)(8z)]^4)`

= `(root(3)(2z))/([2 root(3)(8z)]^4)`

= `1/(2^(23/3).z)`