How can I find the limit of the integral?

[SunGirl]

Homework Statement

Hi! I need to find the limit when x-> +infinity of (integral from x to x^2 of (sqrt(t^3+1)dt))/x^5

Homework Equations

The Attempt at a Solution

The integral of (sqrt(t^3+1)dt) can only be estimated, so sqrt(t^3+1)=(t^(3/2))*sqrt(1+1/t^3) should I use the maclaurin series first for the function sqrt(1 + 1/t^3) (but f`(0) = infinity and I also can`t use maclaurin series for sqrt(t^3+1) as t is infinity) and then take integral for the first several elements? please help, I don`t understand how should I solve this problem.
P.S. Sorry for my bad English :/

 

[Dick]

Homework Statement

Hi! I need to find the limit when x-> +infinity of (integral from x to x^2 of (sqrt(t^3+1)dt))/x^5

Homework Equations

The Attempt at a Solution

The integral of (sqrt(t^3+1)dt) can only be estimated, so sqrt(t^3+1)=(t^(3/2))*sqrt(1+1/t^3) should I use the maclaurin series first for the function sqrt(1 + 1/t^3) (but f`(0) = infinity and I also can`t use maclaurin series for sqrt(t^3+1) as t is infinity) and then take integral for the first several elements? please help, I don`t understand how should I solve this problem.
P.S. Sorry for my bad English :/

Have you thought about trying to apply l'Hopital's theorem?

 

[Gigaz]

You need neither Maclaurin nor l'Hopital, this question is much simpler.
For very large x, the integral must be essentially the integral over t^(3/2) (Though you may want to find a solid reasoning for that, for example with a Taylor series). The rest is very simple.

 

[SunGirl]

You need neither Maclaurin nor l'Hopital, this question is much simpler.
For very large x, the integral must be essentially the integral over t^(3/2) (Though you may want to find a solid reasoning for that, for example with a Taylor series). The rest is very simple.

Thank you very much!) Such a simple solution)