**Definition:** Let be a function defined on the entire number line. The number is called the limit of the function at if for any there exists a number that for all satisfying the condition , the following inequality is satisfied

If , that is, for large (absolute) values of the number very little different from the number 0

If the behavior is different with and separately consider (in the definition of take ) and (definition of take )

## The limit of a sequence

Since a sequence is a function of natural argument , the definition of limit of a sequence with is identical to the definition of limit of a function at

**Definition:** a Number is called the limit of a sequence if for any there is such number that for all , the following inequality is satisfied i.e.

If ,

## Comparison of exponential growth, exponential and logarithmic functions

,

that is

If , when a function grows faster than any exponential function where is a natural number

Graphically, this statement means that **for sufficiently large values of the graph of the function (where ) is above the graph of a function **

,

that is

,

At large ;

,

so

If , the function increases slower than the function (and especially slower than a function or a function )

Graphically, this statement means that **for sufficiently large values of the graph of the function lies below the graph of a function (and especially below the graphs of functions )**