# How do you find the Limit of #n^( ln( (n+1)/n )# as n approaches infinity?

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To find the limit of n^(ln((n+1)/n)) as n approaches infinity, we can use the properties of logarithms and exponential functions.

First, we can simplify the expression inside the logarithm by dividing (n+1) by n:

ln((n+1)/n) = ln(n+1) - ln(n)

Next, we substitute this simplified expression back into the original equation:

n^(ln((n+1)/n)) = n^(ln(n+1) - ln(n))

Using the property of logarithms, we can rewrite this as:

n^(ln(n+1) - ln(n)) = n^(ln(n+1)) / n^(ln(n))

Now, let's consider the behavior of each term as n approaches infinity:

As n approaches infinity, ln(n+1) and ln(n) both approach infinity, but ln(n+1) grows slightly faster.

Similarly, as n approaches infinity, n^(ln(n+1)) and n^(ln(n)) both approach infinity, but n^(ln(n+1)) grows significantly faster.

Therefore, the limit of n^(ln((n+1)/n)) as n approaches infinity is infinity.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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