# Limit of a Sequence

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

a1= 1.

a2= 0.5.

a1000= 0.001.

a1000 000 = 0.000001.

The limit is 0.

a1= 0.5

a2= 0.6666....

a1000= 0.999000999001

a1000 000 = 0.999999000001

The limit is 1.

a1= 5

a2= 7

a1000= 2,003

a1000 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, an, has a limit, L, if and only if for any positive number, ε, there is a term, ak, from which all terms of an greater than ak fulfill that |an−L| < ε.

The limit of the sequence an = 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 a11, the distance to 0 is less than 0.1.

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

From a1001, the distance to 0 is less than 0.001.

## Infinite Limit of a Sequence

A sequence, an, has a limit of +∞ when for M > 0 there is a term, ak, from which all the terms of an greater than ak fulfill that an> M.

The limit of the sequence an = n² is +∞.

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

If M = 10,000, its square root is 100, therefore, for a101 it will exceed 10,000.

a101= 101² = 10,201

A sequence, an, has a limit of −∞ when for N > 0 there is a term, ak, from which all the terms of an greater than akfulfill that an < −N.

Verify that the limit of the sequence an = −n² is −∞.

−1, −4, −9, −16, −25, −36, −49, ...

If N = 10,000, its square root is 100, therefore, for a101 it will exceed −10,000.

a101= −101² = −10,201