# Properties of Limits

1.The limit, if it exists, is unique.

2.If a sequence, **a _{n}**, has a limit, all the subsequences have the same limit as

**a**.

_{n}3. All convergent sequences are bounded.

4.There are bounded sequences that are not convergent.

5. All the monotone and bounded sequences are convergent.

6.There are convergent sequences that are not monotonous.

## Rules of Limits

**lim (a _{n} + b_{n}) = lim (a_{n}) + lim (b_{n}) **

**lim (a _{n} − b_{n}) = lim (a_{n}) − lim (b_{n}) **

**lim (a _{n} · b_{n}) = lim (a_{n}) · lim (b_{n})**

**lim (a _{n} : b_{n}) = lim (a_{n}) : lim (b_{n})**

**lim k · a _{n} =k · lim a_{n}**

**lim a _{n}^{k} = (lim a_{n})^{k} **

**lim log _{a } a_{n} = log_{a} lim a_{n}**

By applying these rules, the following cases are presented:

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