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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. The best Maths tutors available
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1st lesson free!  4.9 (23 reviews)
Intasar
£42
/h
1st lesson free!  5 (17 reviews)
Matthew
£25
/h
1st lesson free!  4.9 (6 reviews)
Dr. Kritaphat
£49
/h
1st lesson free!  4.9 (11 reviews)
Paolo
£25
/h
1st lesson free!  5 (28 reviews)
Ayush
£60
/h
1st lesson free!  4.9 (9 reviews)
Petar
£27
/h
1st lesson free!  4.9 (11 reviews)
Rajan
£15
/h
1st lesson free!  5 (13 reviews)
Farooq
£35
/h

## 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

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.