# How do you determine whether the sequence #a_n=sqrtn# converges, if so how do you find the limit?

so that:

which proves that:

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To determine whether the sequence ( a_n = \sqrt{n} ) converges, consider the behavior of the sequence as ( n ) approaches infinity.

If the sequence approaches a finite number ( L ) as ( n ) approaches infinity, then the sequence converges, and ( L ) is the limit of the sequence.

To find the limit of the sequence ( a_n = \sqrt{n} ), as ( n ) approaches infinity, you can use the fact that the square root function increases without bound as its input increases without bound. Therefore, the limit of ( a_n ) as ( n ) approaches infinity is infinity. Hence, the sequence ( a_n = \sqrt{n} ) diverges.

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To determine whether the sequence (a_n = \sqrt{n}) converges, you can use the properties of sequences and limits. In this case, since (a_n = \sqrt{n}), as (n) approaches infinity, (\sqrt{n}) will also approach infinity. Therefore, the sequence does not converge.

If you were to find the limit of the sequence as (n) approaches infinity, you would write it as:

[\lim_{n \to \infty} \sqrt{n}]

Since (\sqrt{n}) approaches infinity as (n) approaches infinity, the limit does not exist in the real number system. Therefore, the sequence (a_n = \sqrt{n}) diverges.

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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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