Proving a statement about the limit of a specific function (algebraic approach)
Prove that
Let’s use our outline from the Problem-Solving Strategy:
Let
Choose
This choice of
may appear odd at first glance, but it was obtained by taking a look at our ultimate desired inequality:
This inequality is equivalent to
At this point, the temptation simply to choose
is very strong. Unfortunately, our choice of
must depend on
ε only and no other variable. If we can replace
by a numerical value, our problem can be resolved. This is the place where assuming
comes into play. The choice of
here is arbitrary. We could have just as easily used any other positive number. In some proofs, greater care in this choice may be necessary. Now, since
and
we are able to show that
Consequently,
At this point we realize that we also need
Thus, we choose
Assume
Thus,
Since
we may conclude that
Thus, by subtracting 4 from all parts of the inequality, we obtain
Consequently,
This gives us
You will find that, in general, the more complex a function, the more likely it is that the algebraic approach is the easiest to apply. The algebraic approach is also more useful in proving statements about limits.
Proving limit laws
We now demonstrate how to use the epsilon-delta definition of a limit to construct a rigorous proof of one of the limit laws. The
triangle inequality is used at a key point of the proof, so we first review this key property of absolute value.
Definition
The
triangle inequality states that if
a and
b are any real numbers, then
Proof
We prove the following limit law: If
and
then
Let
Choose
so that if
then
Choose
so that if
then
Choose
Assume
Thus,
Hence,
□
We now explore what it means for a limit not to exist. The limit
does not exist if there is no real number
L for which
Thus, for all real numbers
L ,
To understand what this means, we look at each part of the definition of
together with its opposite. A translation of the definition is given in
[link] .
Translation of the definition of
And its opposite
Definition
Opposite
1. For every
1. There exists
so that
2. there exists a
so that
2. for every
3. if
then
3. There is an
x satisfying
so that
Finally, we may state what it means for a limit not to exist. The limit
does not exist if for every real number
L , there exists a real number
so that for all
there is an
x satisfying
so that
Let’s apply this in
[link] to show that a limit does not exist.
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,