Solve this limit when x tends to +infinity

[mohlam12]

please
any hints to solve this limit when x tends to +infinity is way very appreciated !
PS: i should not use the hopital rule...
I tried to factorize the x from the nominator and denominator but couldn't get to any result... i tried some other things.. but still nothing.

$$\frac{x^{\frac{2}{3}} - 3^{x}}{x^{\frac{5}{2}} + 2^{x}}$$

thanks very much

 

[arildno]

Rewrite this as:
$$(\frac{3}{2})^{x}\frac{(\frac{x^{\frac{2}{3}}}{3^{x}})-1}{(\frac{x^{\frac{5}{2}}}{2^{x}})+1}$$

 

[mohlam12]

okay the limit of (3/2)^x is +infinity
but i have to show that the limit of $$\frac{x^{2/3}}{3^x}$$ is zero... how ?? maybe I have to show that it is smaller than a number, then the limit of that number should be zero... by the way, we haven't studied exponentials yet..

PS: I think this should be moved to calculus and beyond ?

 

[Office_Shredder]

the limit of $$\frac{x^{2/3}}{3^x}$$ goes to zero.

EDIT: Latex is so texy

 

[mohlam12]

yes.. but it is an indeterminate form... how is it equal to zero