Function is the limit
Defined Functions limit
Function when the argument approaches a finite limit values
Definition 1
(Continued)
We point out that the definition \ (0 <| x-x_0 | \) represents \ (the X-\ not = x_0 \) , so \ (x \ to x_0 \) when \ (f (x) \) have no limits, and \ (f (x) \) at the point \ (x_0 \) if there is nothing to define.
Definition 1 can be simply expressed as \ (\ lim_ {X \ to x_0} F (X) = A \ Leftrightarrow \ FORALL \ varepsilon> 0, \ EXISTS \ Delta> 0 \) , when the \ (0 <| x-x_0 | <\ delta \) when there is \ (| F (X) -A | <\ varepsilon \) .
Function \ (f (x) \) when \ (x \ to x_0 \) Limiting as \ (A \) is set free to Baidu explanation, here is not expanded.
Arguments approaching the limit function at infinity
If \ (x \ to \ infty \ ) process, a corresponding function value \ (f (x) \) infinitely close to the value determined \ (A \) , then the \ (A \) is called a function \ ( f (x) \) when \ (x \ to \ infty \ ) Limiting.
Definition 2
Provided function \ (f (x) \) when \ (| X | \) is defined greater than the number if there is a positive constant. \ (A \) , for any given positive number \ (\ varepsilon \) ( no matter how small it), there is always a positive number \ (X-\) , such that when the \ (X \) satisfy an inequality \ (| x |> X \ ) , the corresponding function value \ (f (x) \) are satisfies the inequality $$ | f (x) -A | <\ varepsilon, $$ then constant \ (a \) is called the function \ (f (x) \) when \ (x \ to \ infty \ ) Limiting , referred to as \ (\ lim_ {x \ to \ infty} f (x) = A \) or \ (F (X) \ to A \) (when \ (X \ to \ infty \) ).
Can be simply expressed as \ (\ lim_ {X \ to \ infty} F (X) = A \ Leftrightarrow \ FORALL \ varepsilon> 0, \ EXISTS X-> 0 \) , when the \ (| x |> X \ ) , the there \ (| F (X) -A | <\ varepsilon \) .
If \ (x> 0 \) and unlimited increase (denoted \ (the X-\ to + \ infty \) ), so long as the above definition \ (| x |> X \ ) into \ (x> X \ ) , can be obtained \ = a \) (f ( x \ lim_ {to + \ infty x \}) is defined the same way can be obtained. \ (X <0 \) and \ (| X | \) increases infinitely definition of when.