What is the Limit of (4^x + 7^x)^(1/x) as x Approaches Infinity?

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Homework Help Overview

The discussion revolves around finding the limit of the expression (4^x + 7^x)^(1/x) as x approaches infinity, which falls under the topic of limits in calculus.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the behavior of the terms 4^x and 7^x as x increases, questioning which term will dominate. There are attempts to apply logarithmic properties to simplify the limit, with some uncertainty about the implications of taking the natural logarithm.

Discussion Status

The conversation is ongoing, with participants exploring different approaches to the limit. Some guidance has been offered regarding the dominance of terms and the use of logarithms, but no consensus has been reached on the final answer.

Contextual Notes

There is a mention of confusion regarding the application of logarithms and whether it affects the limit, indicating a need for clarity on the rules of limits and logarithmic functions.

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Homework Statement



lim n->infinity (4^x + 7^x)^(1/x)

Homework Equations



Not applicable (?)

The Attempt at a Solution



Tried taking "plugging" infinity for n but I keep getting 1, is this right?

Seems too easy to be just 1?

Thank you.
 
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1 isn't right.

Look at the thing being raised to the power 1/x. Which one will dominate as x gets large?
 
Take the natural logarithm
 
@olivermsun: The 4^x and 7^x will be bigger, but I am lost at the next step.

@flyingpig: Am I allowed to take the natural log of the function? Won't that change the limit?
 
I think olivermsun is saying that one of those two terms will dominate as x gets larger.
 
Find the limit of lnf(x), and then you can find the original limit as [tex]e^{limlnf(x)}[/tex]. Seems 7 to be the final answer.
 
Ah thank you very much Delta, totally forgot about that part :)
 

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