Find the Limit of X and Y Exponents as n Goes to Infinity

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Homework Help Overview

The problem involves finding the limit of an expression involving exponents as n approaches infinity, specifically focusing on the relationship between two variables, X and Y, under the condition that X is greater than or equal to Y and both are positive.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the application of limits and the behavior of the ratio of Y to X as n increases. There are attempts to manipulate the expression by factoring out X^n and considering the implications of the inequality between X and Y.

Discussion Status

Some participants have offered guidance on how to approach the limit, including suggestions to factor out terms and reconsider the placement of constants within the expression. There is an acknowledgment of a mistake in the manipulation of the expression, and while some participants express confidence in the limit being X, there is no explicit consensus on the final outcome.

Contextual Notes

Participants are working under the assumption that X and Y are both positive and that X is greater than or equal to Y, which influences their reasoning about the limit.

jinbaw
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Homework Statement



[tex]X \geq Y > 0, find \lim_{n \to \infty} \left(\frac{2X^n + 7Y^n}{2}\right)^{1/n}[/tex]

Homework Equations





The Attempt at a Solution


I'm not really sure how to do it, but i guess I need to use the fact that [tex]\frac{Y}{X} \leq 1[/tex], and so [tex]\lim_{n \to \infty} \left(\frac{Y}{X}\right)^n = 0.[/tex]
Thanks for your help.
 
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Multiply the argument by [tex]\frac{X^n}{X^n}[/tex] then take the [tex]X^n[/tex] out of the brackets.
 
[tex]\lim_{n \to \infty} X \left( 1 + \left( \frac{7Y}{2X} \right)^n \right)^{1/n}[/tex]
But I'm still stuck on how to procceed, if you could help.
 
There is one mistake, 7/2 should not be within the brackets. Now take X in front of the limit and use [tex]\frac{Y}{X} \leq 1 \Rightarrow \lim_{n \to \infty} \left(\frac{Y}{X}\right)^n = 0[/tex].
 
Oh right.. So its limit is X.
Thanks :-D
 
That's right!
 

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