Limit of a function at a point. Given a function f defined in a reduced neighborhood V of the point x 0 , f is said to have a limit L, when x tends towards x 0 , if whatever is the sequence {x 0 } of points in the neighborhood V that converges towards x 0 , the sequence of the images {f (x n )} converges towards L.
Informally, the fact that a function f has a limit L at point c , means that the value of f can be as close to Las desired, taking points close enough to c , regardless of what happens at c .
Summary
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- 1 Demonstration
- 2 Example on sequence convergence
- 3 Geometric interpretation
- 1 Geometric definition
- 4 Source
Demonstration
To study the behavior of a Function in the vicinity of a given point , considering a function f, defined in a reduced neighborhood V of the point x 0 : By taking a sequence of points of this neighborhood that converges towards x 0 . Let this sequence be said: To each x n of this sequence, its image f (x n ) is associated and thus the sequence of the images can be formed: Therefore, the conclusion is reached that if for any sequence of points in the neighborhood V, that converges towards x 0
, the sequence of the images converges towards the same number L, then we will say that L is the limit of the function f, when x tends towards x 0 .
Example in the convergence of sequences
Considering a constant function defined by f (x) = c and let x 0 be an arbitrary point . So the limit of f when x tends to x 0 is c.
Indeed, if {x n } is a sequence that converges towards x 0 , {f (x n )} is the sequence with the nth term f (x n ) = c, which obviously converges towards c, so the limit of f when x tends to x 0 is c.
Geometric interpretation
From the analysis of these functions, the intuitive idea can be extracted that the limit of a function f, when x tends to x 0 , is L if f (x) can be made to be as close to L as desired, provided that values of x close enough to x 0 . This means that the distance between f (x) and L can be made as small as desired and hence for each positive number £, however small it may be, we must: | f (x) – L | <£ “for certain values of x”.
It can be concluded that for each £> 0 a number ð> 0 must be found in such a way that for all x satisfies
0 <| x – x 0 | <ð to have | f (x) – L | <£. If for all £> 0 this number ð> 0 can be found, it will be said that the limit of the function f when x tends to x 0 is L.
Geometric definition
Given a Function f defined in a reduced neighborhood of the point x 0 , f is said to have a limit L, when x tends towards x 0 , if for every positive number £, there exists a positive number ð, such that: if 0 <| x – x 0 | <ð then | f (x) – L | <£
The two previous definitions of the limit of a Function are equivalent. In case any of them is satisfied, we will say that the limit of a Function exists when x tends to x 0 .
