Question

`lim_(x rightarrow a) ((a + 2x)^(1/3) - (3x)^(1/3))/((3a + x)^(1/3) - (4x)^(1/3)) (a ≠ 0)` is equal to ______.

विकल्प

• `(2/3)^(4/3)`

• `(2/3)(2/9)^(1/3)`

• `(2/9)^(4/3)`

• `(2/9)(2/3)^(1/3)`

Answer 1

`lim_(x rightarrow a) ((a + 2x)^(1/3) - (3x)^(1/3))/((3a + x)^(1/3) - (4x)^(1/3)) (a ≠ 0)` is equal to `underlinebb((2/3)(2/9)^(1/3))`.

Explanation:

`lim_(x rightarrow a) ((a + 2x)^(1/3) - (3x)^(1/3))/((3a + x)^(1/3) - (4x)^(1/3))`  ...`[0/0 "case"]`

Apply L'Hospital rule

`lim_(x rightarrow a) (1/3(a + 2x)^(-2//3). 2 - 1/3. (3x)^(-2//3). 3)/(1/3(3a + x)^(-2//3). - 1/3 (4x)^(-2//3). 4)`

= `(1/3(3a)^(-2//3). (2 - 3))/(1/3(4a)^(-2//3). (1 - 4))`

= `3^(-2//3)/4^(-2//3). 1/3`

= `2^(4//3)/9^(1//3). 1/3`

= `2/3. (2/9)^(1//3)`