Limit of function at infinity

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

  • When
  • ,

    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

  • When
  • ,

    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 )

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