Question

`lim_(n->oo)root(n)(n^2 (sqrt[2])^n + (pi/2)^n + 3 (3/2)^n) =` ?

Answer 1

`pi/2`

Explanation:

We have

`sqrt2 < 1.5 < pi/2` then

`root(n)(n^2 (sqrt[2])^n + (pi/2)^n + 3 (3/2)^n) = pi/2 root(n)((2sqrt[2]/pi)^n n^2+ 1 + 3 (3/2 2/pi)^n)`

here

`2sqrt[2]/pi < 1`
`3/pi < 1`

then

`lim_(n->oo)root(n)(n^2 (sqrt[2])^n + (pi/2)^n + 3 (3/2)^n) =`

`=pi/2 root(n)(lim_(n->oo)(2sqrt[2]/pi)^n n^2+ 1 + 3 lim_(n->oo)(3/pi)^n) = pi/2`

because

`lim_(n->oo)(2sqrt[2]/pi)^n n^2=0`
`lim_(n->oo)(3/pi)^n = 0`