Question

What is the limit of `sqrt(x^3-2x^2+1)/[x-1]` as x goes to infinity?

Answer 1

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

Explanation:

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

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

`= absx sqrt(x-2+1/x^2)`

So

`lim_(xrarroo)sqrt(x^3-2x^2+1)/(x-1) = lim_(xrarroo)(absx sqrt(x-2+1/x^2))/(x(1-1/x))`

`= lim_(xrarroo)(sqrt(x-2+1/x^2))/(1-1/x)`

`= oo`