So let us try to generalize the concept of
the limit of the sequence into
the case of functions of real domain.
Basically, what we've actually already have,
we have the idea that the limit is a value of
not just any real value which resembles
our functions or our sequence the most.
The rule here is that if you have any given deviation,
then we can assume that
our sequence or our function does not
deviate much from the limit
if it get close enough to the limit point.
So let us try to formally write this down.
Well, this is nightmarish because we are going to use
our quantifiers as been transduced on
the previous week but everything will be fine.
So let us start from our picture.
What we do have here?
We do have here our function,
this is a blue curve and then have a point
of interest or limit point a here.
So what's an idea?
What I'm going to say is that this function
have a limit equal
to c as x approaches a and c is drawing over there.
So what does it mean?
It basically means that if we just state that we
allow our function to deviate from the limit by Epsilon,
the Epsilon just any possible real positive value,
then we create some boundaries
on our functions were derived,
c plus Epsilon and c minus epsilon.
So this is our epsilon tube
here and the idea is that we can
get as close to point a as
the function doesn't break our Epsilon tube.
In our case, we've come up with the idea of Delta here,
neighborhood of the point a that's
Epsilon tube and Delta tube intersects in rectangle and
our function wholly lies within the rectangle, that nice.
So the second thing we need to do
is to change our Epsilon because what we
want achieve we want to achieve the ideas that we can do
this procedure for all possible deviations
so that's how our definition rock.
It's extremely formal one
but we should always start with the formal things.
So for all deviations,
we can get as close to our limits
point thus function does not deviate
from its linked much than previously stated division.
That's how it basically can
be read in natural language, that's fine.
But there is bomer here,
why do we need to write that
our absolute value of x minus a or
this the distance between
arbitrary point in the neighborhood of the
a is smaller than
Delta which basically means at its neighborhood.
But what is greater than 0,
greater than 0 actually means
that we are not considering as appoint a itself.
So basically, it means that we are looking at
every where there is a function as neighborhood
except the point itself and twice it.
Let us assume that there is some enemy around
us and this enemy come at the very point a
in just damaged our functions and served that f from
a for example is not
how nice easy blue curve
and sample as a point for example this one.
Remember our definition, well
hand-waving definition so what we've said previously,
we've said that the limit is the value that
resembles our function in the neighborhood.
As we get as close to the limit point,
we are not supposed to be ends on this point.
We are considering only functions in
the neighborhood of it and
the neighborhood function doesn't
actually care about the value of the function at
the very point itself so we should actually
expel our x equal to a from
zircons duration of the limit, so it's tricky.
Sometimes people will use
those ideas that we need to have an as
more sequence-based definition for
the limit function and it's
expectable because it does write in the following manner.
If we assume any sequence of
arguments of our function x_n which approach
is our limit point n which can be done
because a is our limit point,
then the sequence of function values on this sequence of
functions hiker events should
necessarily approach c, approach it's limit.
It nice because it operates
in the same joint set that we've already have been
established as a limit of
a sequence but the problem here is that
we're talking about all possible approaches
to the point a,
all possible sequences that approaches
point a which nightmarish object.
You can even just imagine
how much sequences approaches point a.
So the first definition is usually used to
prove that there is a limit
and to calculate it and the second definition
is used to prove that
there is no limits or this point is not really much.
So it used to prove an absence of a limit.