# Limit of a Sequence

The limit of a sequence is the number which the terms of a sequence are approaching.

a_{1}= 1.

a_{2}= 0.5.

a_{1000}= 0.001.

a_{1000 000 }= 0.000001.

The limit is **0**.

a_{1}= 0.5

a_{2}= 0.6666....

a_{1000}= 0.999000999001

a_{1000 000 }= 0.999999000001

The limit is **1**.

a_{1}= 5

a_{2}= 7

a_{1000}= 2,003

a_{1000 000 }= 2,000,003

No particular number can represent the limit of this sequence, therefore, the limit is **∞**.

## Finite Limit of a Sequence

A squence, **a _{n}**, has a

**limit, L,**if and only if for any positive number,

**ε**, there is a term,

**a**, from which all terms of

_{k}**a**greater than

_{n}**a**fulfill that

_{k}**|a**.

_{n}−L| < ε The limit of the sequence a_{n} = 1/n is 0.

It can be determined from that term of the sequence that the distance from 0 is less than a positive number (**ε**).

From **a _{11}**, the distance to 0 is less than 0.1.

Determine from that term if the distance to 0 is less than 0.001.

From **a _{1001}**, the distance to 0 is less than 0.001.

## Infinite Limit of a Sequence

A sequence, **a _{n}**, has a

**limit of +∞**when for

**M > 0**there is a term,

**a**, from which all the terms of

_{k}**a**greater than

_{n}**a**fulfill that

_{k}**a**.

_{n}> M The limit of the sequence a_{n} = n^{2} is +∞.

1, 4, 9, 16, 25, 36, 49, ...

If M = 10,000, its square root is 100, therefore, for **a _{101}** it will exceed 10,000.

a_{101}= 101^{2} = 10,201

A sequence, **a _{n}**, has a limit of

**−∞**when for

**N > 0**there is a term,

**a**, from which all the terms of

_{k}**a**greater than

_{n}**a**fulfill that

_{k}**a**.

_{n}< −N Verify that the limit of the sequence **a _{n} = −n^{2} is −∞.**

−1, −4, −9, −16, −25, −36, −49, ...

If ** N = 10,000**, its square root is 100, therefore, for ** a _{101}** it will exceed −10,000.

a_{101}= −101^{2} = −10,201