Question

How do you find the limit `lim (root4(x+1)-root4x)x^(3/4)` as `x->oo`?

Answer 1

`1/4`

Explanation:

`lim_(x to oo) (root4(x+1)-root4x)x^(3/4)`

`= lim_(x to oo) x^(1/4)(root4(1+1/x))-1)x^(3/4)`

`= lim_(x to oo) x ( root4(1+1/x)-1 )`

By Biniomial Expansion:

`= lim_(x to oo) x((1+1/(4x) + O(1/x)^2)-1 )`

`= lim_(x to oo) 1/(4) + O(1/x)^1 = 1/4`

Answer 2

`1/4`

Explanation:

`root(4)(x+1)-root(4)(x)=(root(4)((x+1)^2)-root(4)(x^2))/(root(4)(x+1)+root(4)(x)) = (root(4)((x+1)^4)-root(4)(x^4))/((root(4)(x+1)+root(4)(x))(root(4)((x+1)^2)+root(4)(x^2)))=1/((root(4)(x+1)+root(4)(x))(root(4)((x+1)^2)+root(4)(x^2)))=1/(x^(3/4)(root(4)(1+1/x)+root(4)(1))(root(4)((1+1/x)^2)+root(4)(1)))`

then

`lim_(x->oo)(root(4)(x+1)-root(4)(x))x^(3/4) = 1/4`