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

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The limit of (4^x + 7^x)^(1/x) as x approaches infinity is determined to be 7. As x increases, the term 7^x dominates over 4^x, leading to the conclusion that the limit simplifies to e^(lim ln(4^x + 7^x) / x). The natural logarithm is utilized to analyze the limit, confirming that the final answer is indeed 7.

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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 e^{limlnf(x)}. Seems 7 to be the final answer.
 
Ah thank you very much Delta, totally forgot about that part :)
 

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