The limit of a sequence

Definition 11. A sequence is an infinite set of terms, each of which is assigned a number. The terms of a sequence must obey a certain law.

Example.

Definition 12. A number is called the limit of a sequence if, for any e>0, there exists a number N(e) depending on e, such for .

Notation: .

Example.

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Definition 13. The limit of a variable х is a number а such that for any e>0, there exists an х starting with which all х satisfy the inequalities .

Properties:

1. The limit of a constant number equals this number.

2. A variable can not have two different limits.

3. Some variables have no limit.

Definition 14. We saythat х tends to infinity if, for any number М, there exists an х such that, starting it, .

(a) M> 0, x>M, x ®+¥;

(b) M< 0, x<–M, x ® –¥.

Definition 15. A number b is called the limit of the function f(x) as х® а if, for any given e>0, there exists a small positive d depending on e (d(e)>0) such that, for any х satisfying the inequality , . Notation: .

Definition 16. The left limit of a function f(x) as x® a is the limit of f(x) as x® a, and х<а. Notation: .

Definition 17. The right limit of a function f(x) as x® a, is the limit of f(x) as x® a, and х>а. Notation: .

If the left limit equals the right limit and some number b, then b is the limit of the function as x® a.

The denominator of the last fraction is very small, and the fraction is very large: . Decreasing the denominator and, finally, divide 1 by 0, we obtain infinity, i.e.,

Using this relation, we find , .

Definition 18. A number b is called the limit of f(x) as х®¥ if, for any e>0, there exists a (large) number N depending on e such that for any .

Notation: .

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