What is the limit of the expression (3sqrt{n})^(1/2n)?

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In summary, the limit of (3 sqrt(n))^(1/2n) is equal to 1. To calculate the limit, you can use the power rule for limits and rewrite the expression as (3^(1/2n)) * (n^(1/4n)). The limit is significant in understanding the behavior of a function as the input (n) approaches infinity. There are various real-life applications for this expression, but there are also limitations to consider when using it in calculations.
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shannon
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Homework Statement


Determine the limit of:

lim ((3sqrt{n})^(1/2n))


Homework Equations





The Attempt at a Solution


I don't even know where to begin...perhaps squaring the entire term so I get..

9n^(1/[4n^2]) which is equivalent to n^[1/(n^2)]

But I don't know what the limit of that is...I graph it and get all kinds of craziness
Please help!
 
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The limit as n -> ?
 

What is the limit of (3 sqrt(n))^(1/2n)?

The limit of (3 sqrt(n))^(1/2n) is equal to 1.

How do you calculate the limit of (3 sqrt(n))^(1/2n)?

To calculate the limit of (3 sqrt(n))^(1/2n), you can use the power rule for limits and rewrite the expression as (3^(1/2n)) * (n^(1/4n)). Then, you can take the limit of each term separately to get the final result of 1.

What is the significance of the limit of (3 sqrt(n))^(1/2n)?

The limit of (3 sqrt(n))^(1/2n) is important in mathematics because it helps us understand the behavior of a function as the input (n) approaches infinity. In this case, we see that as n gets larger and larger, the expression approaches a constant value of 1.

What are some real-life applications of (3 sqrt(n))^(1/2n)?

This expression can be used in various fields such as finance, physics, and engineering. For example, it can be used to model the growth of a population or the decay of a radioactive substance.

Are there any limitations to using (3 sqrt(n))^(1/2n) in calculations?

Yes, there are some limitations to using (3 sqrt(n))^(1/2n) in calculations. This expression is only valid for non-negative values of n and may not accurately represent the behavior of a function for all values of n. Additionally, there may be other factors that affect the growth or decay of a system that are not accounted for in this expression.

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