The limit of a sequence.

Function. function limit. Fundamental theorems on limits. Infinitely small and infinitely large quantities. The ends

The Theory of Limits

The limit of a sequence.

Definition.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

х₁,х₂,х₃,х₄,…,х_n,….

Example.

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

for n>N,

Notation:

.

Example.

.

Example. . Assigning integer values to n, we obtain: etc.; i.e. this variable has no limit.

The limit of a function.Suppose that y=f(x) is a function defined on a domain D containing a point а.

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

. (1)

Example. Find the limit of f(x)=5x–1 as x→2

,

,

i.e., .

To find δ, we must find x from the inequality for the function and substitute it in the inequality for the variable.

Definition.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.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.

Definition.A number b is called the limit of f(x) as х→∞ if, for any ε >0, there exists a (large) number N depending on εsuch that for any .

Notation:

.

Definition.A function f(x) is said to be infinitesimals as х→ а if, for any М, there exists a δsuch that whenever .

Notation:

.

Definition 1. A function α (х) is called an infinitesimal as х→ а if

.

Theorem. The sum of finitely many infinitesimals is an infinitesimal:

α₁(х) + α ₂(х) + α ₃(х) + … + α _к(х)= α (х).

Theorem. The product α (x)^.z(x) of an infinitesimal α (х) by a bounded function z(x) as x→a is an infinitesimal.

Corollary.The product of infinitesimals is an even smaller quantity.

Theorem. An infinitesimal divided by a function having nonzero limit as х→ а is infinitesimal, i.e., if

, then is an infinitesimal.