Solve this limit when x tends to +infinity

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Homework Help Overview

The discussion revolves around evaluating a limit as x approaches positive infinity, specifically the expression \(\frac{x^{\frac{2}{3}} - 3^{x}}{x^{\frac{5}{2}} + 2^{x}}\). Participants are exploring various approaches to solve this limit without using L'Hôpital's rule.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to factor out x from the numerator and denominator but struggles to find a result. Another participant suggests rewriting the limit in a different form to analyze its components. There is a discussion about showing that \(\frac{x^{2/3}}{3^x}\) approaches zero, with questions about how to demonstrate this without prior knowledge of exponentials.

Discussion Status

Participants are actively engaging with the problem, offering hints and alternative forms of the limit. There is recognition of the indeterminate nature of the limit, and some participants are questioning how to justify certain limits without having studied relevant concepts in depth. No consensus has been reached yet.

Contextual Notes

There is a mention of imposed homework rules, specifically the restriction against using L'Hôpital's rule, which may limit the approaches available to participants. Additionally, one participant notes that they have not yet studied exponentials, which may affect their understanding of the problem.

mohlam12
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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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