Prove that function hasn't lim in #x_0=0# ?
Function:
#f(x)={(2 " when " x=(1,1/2,1/4...)),
(1 " when " x=R\(1,1/2,1/4...)):}#
Function:
2 Answers
See explanation.
Explanation:
According to Heine's definition of a function limit we have:
So to show that a function has NO limit at
and
In the given example such sequences can be:
Both sequences converge to
because all elements in
and for
but for all
So for
Both sequences coverge to
QED
The limit definition can be found in Wikipedia at: https://en.wikipedia.org/wiki/Limit_of_a_function
Here is a proof using the negation of the definition of the existence of a limit.
Explanation:
Short version
So no matter what someone proposes for
Long version
there is a number,
The negation of this is:
for every number,
Given a number
Now given a positive
Given a positive
There is also an element
If
If