Question

Question #6fabd

Answer 1

`-3/2`

Explanation:

From

`(x-1)^n = x^n + (n x^(n-1)(-1))/(1!)+(n(n-1)x^(n-2)(-1)^2)/(2!)+ cdots +`
Considering `n = 3/2` we have

`(x-1)^(3/2) - x^(3/2) = (3/2 x^(3/2-1)(-1))/(1!)+(3/2(3/2-1)x^(3/2-2)(-1)^2)/(2!)+cdots+` and

`((x-1)^(3/2)-x^(3/2))/sqrt(x) = (3/2 x^(3/2-1-1/2)(-1))/(1!)+(3/2(3/2-1)x^(3/2-2-1/2)(-1)^2)/(2!)+cdots+=`

`(3/2 x^0(-1))/(1!)+x^-1 f(x^-1)` then

`lim_{x\to\infty}((x-1)^(3/2)-x^(3/2))/sqrt(x) =lim_(x->oo) (3/2 x^0(-1))/(1!)+x^-1 f(x^-1) = -3/2`

Answer 2

`lim_(x->oo)((x-1)^(3/2)-x^(3/2))/sqrt(x)=-3/2`

Explanation:

`lim_(x->oo)((x-1)^(3/2)-x^(3/2))/sqrt(x)`
`=lim_(x->oo)(((x-1)^(3/2)-x^(3/2))((x-1)^(3/2)+x^(3/2)))/(sqrt(x)((x-1)^(3/2)+x^(3/2)))=`
`[`using `(a-b)(a+b)=a^2-b^2]`
`=lim_(x->oo)((x-1)^3-x^3)/(sqrt(x)((x-1)^(3/2)+x^(3/2)))`
`=lim_(x->oo)(x^3-3x^2+3x-1-x^3)/(sqrt(x)(x-1)^(3/2)+sqrt(x)x^(3/2))`
`=lim_(x->oo)(-3x^2+3x-1)/(sqrt(x)sqrt(x-1)(x-1)+x^2)`
`=lim_(x->oo)(-3+3/x-1/x^2)/(sqrt(1)sqrt(1-1/x)(1-1/x)+1)=(-3)/(1+1)=-3/2`