Limit of a function in a point
A very similar notion to the notion of limit of a sequence is the notion of limit of a function in a point. It describes the behaviour of values of the function when the arguments are nearer and nearer to a given number. There are two equivalent definitions of this notion: due to Heine and to Cauchy.
We will say that a function has at a point limit (denoted as ), if for any sequence which converges to and such that for any , , the limit of the sequence exists and equals .
Simple example: function has at limit , because from the arithmetic of limits of sequences we know that if converges to , then converges to .
On the other hand, function
has no limit at , because for sequence , , and for , .
The equivalent definition is the following. We will say that a function has limit at point , if:
which means that for arbitrarily positive small number there exists a small interval around , such that values of the function in this small interval differ from the limit no more than .
Let us check, that . Let . It suffices to set . Then, if , then actually .
Infinite limits and limits at infinity
Using Heine’s definition we see that we can easily consider also limits of a function if lub . Simply we can take all sequences , which converge respectively to or and look at the limits of sequences . So for example, .
Also it can occur that the limit of a function at a given point is or , if for all sequences are such that the limit of is respectively or . E.g. }.
Arithmetic of limits
The Heine’s version of definition immediately implies that the arithmetic of limits of functions works in the same way as the arithmetic of limits of sequences. E.g. .
Left limit (from below) of a function (denoted as ) equals , if for any sequence which converges to , such that for any , we get . Similarly we define right limit (from above) (denoted as ) equals , if for any sequence which converges to , such that for any , we get
E.g. and .
A function has a limit at if and only if it has both limits in those points and they are equal.
We have the following theorem: if and , if for some neighbourhood of , for .
Sounds a bit complicated but is very convenient. E.g. let us calculate the limit of function for . Let . If , then , and if only . Therefore:
Asymptotes are lines which are lines to which the diagram of a function converges. Asymptotes can be vertical, horizontal or oblique.
If , then line is a right vertical asymptote. Analogously, if , this line is a left vertical asymptote. E.g. oraz , so asymptote is a vertical asymptote of this function.
If or , then line is a respectively right or left horizontal asymptote of this function. Since , is a horizontal asymptote of .
A line is an oblique asymptote (respectively left or right), if or . If such a line is asymptote (assume it is a right asymptote), then oraz .
E.g.: let , then and . Therefore, is an oblique asymptote of this function.
A function is continuous at a point , if the limit exists and equals . Obviously all simple arithmetic functions are continuous in every point of their domains. E.g. if , .
A function which is continuous at every point of its domain will be simply called continuous.
Function is continuous. Obviously it is continuous for all non-zero points. It is also continuous for , because and , and therefore .
Let be a function defined in the following way:
This function is continuous in all points except , and is not continuous at , because .
Let function be defined as follows:
This function is continuous for , but is not continuous in , because , but , and therefore it has no limit at point .
Function defined in the following way:
is an example of function which has no limit in any point. Therefore is also not continuous at any point.
Continuous function on an interval have the following (intuitively obvious) property: if , then there exists , such that , and . Therefore, for example, since for is continuous and and , and therefore has at least one root in the interval .
If we are able to choose for each , in a such way that in any interval of length values of functions do not differ more than , universally regardless of place , then we say that the function if uniformly continuous. More formally a function is uniformly continuous, if:
Function is a very simple example. It is uniformly continuous, because for any set . Then for all such that , then obviously .
On the other hand, function on the interval is continuous but is not uniformly continuous, because if , regardless how small we choose, we can find , such that , and therefore .