Limits & continuity
The limit concept
The limit is one of more important concepts in calculus for analysis of functions. Definitions such of derivate and integral are based in this important concept.
Shortly formally, the limit of a real variabled function f in a point a, is the value wich tends function when x tend to the point a. It mains the tendence of function f in a neighborhood of x.
Rigorous definition is
Lets f a real variabled function. It is said that the limit of f at \( x_{0} \) is L when
$$ \forall \epsilon>0, \exists \delta > 0 : |x – x_{0}| < \delta \Rightarrow |f(x) – L | < \epsilon .$$
It is denoted as
$$ \lim_{x \mapsto x_{0}} f(x) = L $$
Continuity concept
Continuity concept deals with neighborhood. Shortly formally, continuity means that if two ponits x, y are neighbors then its images through f, f(x) and f(y) are also neighbors.
Limit is related with continuity concept. A function f is continuous at a point \( x_{0} \) following two conditions holds:
- Function f has finite limit L at \( x_{0} \)
- Value of function at \( x_{0} \) is defined and it´s value is L
Both conditions can be written more fomally as follows:
Lets f a real variabled function. It is said that f is continuous at \( x_{0} \) if
$$ \forall \epsilon>0, \exists \delta > 0 : |x – x_{0}| < \delta \Rightarrow |f(x) – f(x_{0}) | < \epsilon .$$
Or equivalently
$$ \lim_{x \mapsto x_{0}} f(x) = f(x_{0}) $$
It is obvious the fact that a function f is continuous at \( x_{0} \) then it has limit at this point. Converse does not holds, ie, a funcion can have limit at a certain point, but not neccesarilly be continuous in such point as showed in following example:
Example 1
Lets f(x) defined as follows.
$$f(x) = \left\{\begin{matrix} 1 & x \neq 0 \\ 0 & x = 0 \end{matrix}\right. $$.
Note that $$ \lim_{x \mapsto 0} f(x) = 1 $$ but \(f(0)=0\)
Example 2
Lets f(x) defined as follows.
$$f(x) = \left\{\begin{matrix} \frac{x^2}{2} & x \leq 1 \\ 1-x & x>1 \end{matrix}\right. $$.
We have found an interactive graphic for this case (you can drag into):
In this case, function is not continuous and nor has limit at x=1.
Note that in this interactive graphic you can drag the x point and also play with the \(\delta\) value. If you drag the point to x=1, for its neighboring points points \( (1-\delta, 1+\delta) \) there not exsits any \(\epsilon\) for wich \( |x-1|<\delta \Rightarrow |f(x) - f(1)|< \epsilon \). In other words, \(\delta\) can be as small as we want, but \(\epsilon \) cannot be smaler that jump of function at x=1.
We want to introduze other limit definitions:
Lets f a real variabled function.
$$ \lim_{x \mapsto \infty }f(x)=L \Leftrightarrow \forall \epsilon> 0, \exists M > 0 : x>M \Rightarrow |f(x) – L | < \epsilon .$$ $$ \lim_{x \mapsto x_{0} }f(x)=\infty \Leftrightarrow \forall M> 0, \exists \delta > 0 : |x-x_{0}|< \delta \Rightarrow f(x) > M .$$
It is analogous to define negative infinite limits
Lets f a real variabled function.
Rigth limit definition:
$$ \lim_{x \mapsto a^{+} }f(x)=L \Leftrightarrow \forall \epsilon> 0, \exists \delta > 0 : x+a <\delta \Rightarrow |f(x) - L | < \epsilon .$$ Left limit definition: $$ \lim_{x \mapsto a^{-} }f(x)=L \Leftrightarrow \forall \epsilon> 0, \exists \delta > 0 : a-x <\delta \Rightarrow |f(x) - L | < \epsilon .$$
There is an important theorem that sometimes it is used as definition of limit. It say that the limit of a function f exists then both lateral limits also exists and are equaly, more formally:
Lets f a real variabled function. If
$$ \exists L : \lim_{x \mapsto a }f(x)=L \Leftrightarrow \lim_{x \mapsto a^{-} }f(x)=L=\lim_{x \mapsto a^{+} }f(x) $$
Example
Lets the function
\( f(x)=\frac{1}{x^2}\) then
$$ \lim_{x \mapsto 0 }\frac{1}{x^2}= \infty $$
Funtion has infinite positive limit at x=0.
Example
Lets
\(f(x)=\frac{1}{x}\) then
$$ \lim_{x \mapsto 0 }\frac{1}{x}= ?$$
In fact funtion f, has not limit at x=0. Because right limit when x maps to 0 is infinite but left limit when x maps to 0 is -infinite.
Note that, in this last two examples, lateral limits matching in first case, but no in second case.
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