Question

Limit x tends to infinity ((x+1)(x+2)(x+3))^1/3-x=?

Answer 1

`1`

Explanation:

`(x+1)(x+2)(x+3) = x^3+bx^2+cx+d`

`root(3)((x+1)(x+2)(x+3)) = root(3)(x^3+bx^2+cx+d)`

`= root(3)(x^3(1+b/x+c/x^2+d/x^3))` `" "` for `x != 0`

`= xroot(3)(1+b/x+c/x^2+d/x^3))` `" "` for `x != 0`

So

`lim_(xrarroo)((x+1)(x+2)(x+3))^(1/3)/x = lim_(xrarroo)root(3)(1+b/x+c/x^2+d/x^3)`

`= root(3)(1+0+0+0) = 1`