Limit of a function
In mathematics, the limit or the limit of a function at a particular point refers to that value, which approximates the function in the neighborhood of the point under consideration. However, such a limit value does not exist in all cases. The limit exists, converges the functionality, otherwise it diverges. The limit concept was formalized in the 19th century. It is one of the most important concepts of calculus.
- 3.1 Definition
- 3.2 Examples
- 3.3 notation
- 3.4 One-sided and two-sided limit
- 5.1 derivative and differentiability
Formal definition of the limit of a real function
The symbol is read " the limit of f x for x against p " denotes the limit of the real function for the border crossing of the variables compared. This can be both a real number and one of the symbolic values and. In the first case does not necessarily lie in the domain of, but there must be a limit point of, that is, in every neighborhood of infinitely many elements of must be. In the case or the domain of must be unlimited up and down.
Accordingly, there are several variants of the definition of the limit concept:
Argument finally, finally limit
- Definition: Let be a subset of and a limit point of. The function has the limit, if there is a (generally dependent ), for every ( however small ), so that also applies to all values from the domain of which satisfy the condition.
Expressed qualitative terms the definition: The difference between the function value and the limit is arbitrarily small, if one chooses sufficiently close to.
Note that it does not matter which value assumes the function of the body; The function does not even need to be defined at the point. The only factor is the behavior of the dotted environments. However, some authors use a definition with environments that are not dotted; see the section Newer term limit.
In contrast to the use of Augustin -Louis Cauchy formulation that " the function of the threshold approach ", a variable that is "running" is but just one element of a predetermined amount. This static ε - δ definition used today is mainly due to Karl Weierstrass and introduced the term limit on a firm mathematical foundation, the so-called epsilontics.
Example:
Argument finite, infinite limit
- Definition: The function of ( with ) the limit, if there is a (generally dependent ) are at all ( however large ) real number, so that is also true for arbitrary values from the domain of which satisfy the condition.
According to the case, the limit is defined.
Example:
Argument infinite, finite limit
- Definition: The function has the limit, if there is a (generally dependent ) real number to each ( however small ), so that is also true for arbitrary values from the domain of which satisfy the condition.
Accordingly, limits the type or define leave.
Example:
Definition with the help of consequences
In the real numbers can be an accumulation point characterized as follows:
Let be a subset of and. is a limit point of if and only if it is a sequence that satisfies, see limit ( sequence).
This property can formulate an alternative definition of limit:
- Definition: Let a function which is a limit point of and. Then one defines if and only applies if for every sequence with and.
Once one accepts as a limit in the definition of accumulation point, you might just as well and define.
It can be shown that - the definition of the threshold value is equivalent to the sequence definition.
One-sided limits
Definition
Let be a subset of and a limit point of. The function has the limit, if there is a (generally dependent ), for every ( however small ), so that also applies to all values from the domain of which satisfy the condition.
Accordingly, limits the type or for defined.
Examples
Notation
One-sided and two-sided limit
To avoid confusion, it is called in the case of the both- sometimes limit. If a cluster point of the and is, then: exists if and only if the two -sided limits and exist and coincide. In this case, the equality is true.
Limit theorems
Be, and two real-valued functions whose limits exist and, with and an accumulation point of from the extended real numbers. Then there exist the following limits and can be calculated as indicated:
If, in addition, it also exists, and it is
- .
Applies as well, so it can not apply the limit theorem. In many cases, one can determine the limit but with the rule of L' Hospital.
Additionally, the following Schachtelungssatz applies:
- If and, it is.
Application
The application of the threshold concept to the difference quotient has proved to be particularly fruitful. It forms the very basis of the analysis.
Derivative and differentiability
(Also called leads) differential quotients are the limits of the difference quotient of a function, ie expressions of the form
Where and denote. Spellings such as or, if this limit exists. Use the properties and the calculation of differential coefficients, the differential calculus is concerned.
There is a differential coefficient of a function at the point, then the function is differentiable at the point.
Important limits
The occurring in the derivation of the power function limit can be calculated using the binomial theorem:
Which occurs when the derivative of the exponential limit requires the introduction of the Euler's number, and based thereon the natural logarithm:
The derivation of the trigonometric functions leads ultimately to the limit. For the calculation of this limit, there are different approaches, depending on how the trigonometric functions and the number Pi are defined analytically. If you measure the angle in radians, we obtain
Newer term limit
More recently, a variation of the threshold concept is used, the working with environments that are not punctured. Using consequences of this variation defines the limit as follows: Let be a function, an element of the closed shell and. Then one defines if and only applies if for every sequence with and.
The difference to the above given dotted variant consists, first, that now is no longer prohibited if. Second, by a definition at all points in the closed hull is possible, and thus particularly to isolated points of.
An equivalent non-punctured - definition of the limit can also specify easily: In the above given - definition be replaced by, so also the case be expressly permitted needs.
The non -dotted version is not equivalent to the dotted version. In particular, it differs at points of discontinuity:
In the dotted version is continuous at if and only if the limit exists and is valid for, or if an isolated point is. In the non -dotted version, however, it is enough for continuity to demand the existence of the limit, the equation is thus automatically satisfied.
Example:
This feature is not continuous. The limit in the non -dotted sense does not exist. The limit in the punctured sense exists, however: because it is specifically requested and is valid for these values. Obviously, however.
To avoid misunderstandings recommend the representatives of the non -dash variant, to denote the dotted limit for the following:
The representatives of the newer version to see the advantage of their variation over the classical dotted variant of Weierstrass that can limit theorems with the newer variant easier to formulate because of the special cases that arise by stippling, no longer need to be considered.
Limit of a function with respect to a filter
Both the classical limit notion of Weierstrass and the newer term limit can be seen as special cases of the general concept of limit of a function with respect to a filter interpret:
Let be a function from to, which is provided with a topology, and a filter on. A point is called the threshold function with respect to the filter when the filter generated by the filter converges to the base, so if the filter generated by the filter base is finer than the area of the filter.
The newer definition of the limit of a function at the point now corresponds to the special case that is chosen as the area of the filter; the classical definition of Weierstrass corresponds to the special case that is chosen as the produced by the dotted environments of filter.