Solve this limit when x tends to +infinity

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The discussion focuses on solving the limit of the expression (x^(2/3) - 3^x) / (x^(5/2) + 2^x) as x approaches infinity, specifically without using L'Hôpital's rule. The user has attempted to factor out x but has not achieved a satisfactory result. They rewrite the limit in terms of exponential functions, noting that (3/2)^x tends to infinity, and express a need to show that the limit of (x^(2/3) / 3^x) approaches zero. The conversation highlights the challenge of proving this limit, especially since the user has not yet studied exponential functions in depth.
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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
 
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Rewrite this as:
(\frac{3}{2})^{x}\frac{(\frac{x^{\frac{2}{3}}}{3^{x}})-1}{(\frac{x^{\frac{5}{2}}}{2^{x}})+1}
 
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 ?
 
Last edited:
the limit of \frac{x^{2/3}}{3^x} goes to zero.

EDIT: Latex is so texy :smile:
 
Last edited:
yes.. but it is an indeterminate form... how is it equal to zero
 

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