Since we spoke about it earlier,
let us revisit the concept of indeterminate forms.
Whilst we were talking about limits of sequences,
we established that there are
quite a lot of nightmarish cases.
When we are not able to tell the limit straightaway.
Basically, it was a case as when considering
our limit of our function by
arithmetic rules resulted in some kind of weird result.
For example zero divided by zero,
infinity divided by infinity and so on.
I'm going to test this out one more time.
One power to infinity is also indeterminate form where
one is basically a function that approaches
one not arithmetic number.
Once again, let me remind you that
all these indeterminate forms are kind
of connected because one can be derived from another
and the basic one is a zero divided by zero.
For example, an easy case that we have zero
divided by zero or we have, for example,
one divided by infinity and one
divided by infinity thus we
have infinity divided by infinity.
[inaudible] we just take the inversion of both functions,
that's how we get our infinity divided by infinity.
Same applies for example for a one powered to infinity.
If we can see the transition to
exponential form of which by the definition of logarithm,
then it is exponents
of a logarithm of one powered to infinity.
That's the definition of logarithm.
Thus we have exponent of infinity
multiplied by a logarithm of one which approaches zero.
So we have our zero multiplied by infinity which can be
easily truncated into the form of
infinity divided by infinity or zero divided by zero.
Basically, all the same and have the same roots.
It's kind of nice but
still the general idea is if you remember all of them,
it's much more easy for you to not have
any additional mistakes while calculating limits.
Let us consider some example here.
This is the very obvious thing.
Let us assume that we are calculating the limit of
two polynomial functions where
basically my end coefficients are non-zero.
We have polynomial function of nth degree in
the numerator and mth degree in the denominator.
So what's limit if x approaches infinity?
Basic idea here is that let us for example divide
by x power to m both numerator and denominator.
What do we get? In the numerator,
we get a_0 divided by x power
ten plus a_ one divided by
x power a minus one plus so on, so on.
A minus one divided by x plus a.
If the numerator, we get something a bit more complicated
because if we divide it by x power n
we actually do not know relation between n and
m. Maybe by this very moment,
we have already exceeded all m degrees of x to get,
for example, simple polynomial
function after division or maybe not.
Let us for example consider the case of when m is
greater than n. Then as the result of division,
we get b_0 divided by x_
n and b_1 divided by x_ n minus one,
plus so on, so on.
Here, we get some b, for example,
I am going to say that it's number's k and so on, so on.
After all this division,
they get the last multiplier.
The last term is b_ m multiplied
by x_m minus n. Of course,
as you all don't understand this is true
for all m and n without any order.
But the m minus n positive or negative,
then the limit of the numerator and
the denominator are zero or infinity.
Let us just gradually solve what we do have here.
Firstly, we can easily tackle denominator here because
all these fractions approaches
zero by our arithmetic rule.
Because a zero is
a constant and the
denominator in all these cases approaches infinity
because x approaches infinity.
A constant divided by infinity gets result to zero.
The numerator actually approaches our a_nth.
The same applies for the denominator
right up the case of b_k.
If we are considering b_k,
b_k actually approaches b_k.
If m is greater than n,
then we got a bunch of infinitesimal terms here,
then some finer term and then
some polynomial function on the infinity,
which is obviously infinity.
Thus we got a constant
divided by infinity and as a result,
we get infinity here.
It's easy to understand that if we
just consider another order here,
then we get zero.
Sorry, vice-versa obviously.
We just get that b in case if m greater than n,
we get constant divided by
infinity thus we get zero and for our case,
we get infinity divided by a constant which is infinity.
That's quite nice. But what
about the case where m equals to n.
That's quite easy because that basically means that
after our division by x_ n,
we get only our main coefficients here and the answer
is a_m divided by a b-m
or b_n since n equals to m. That basically covers
all possible relations between
two polynomial functions with infinite argument here.
We've just resolved the indeterminate form
for the case of infinity divided by infinity.
Let us also assume one of the cases
that we've actually seen already for the sequences.
Remember, we have considered sequence sine of
n divided by n. We spoke
about that a sine is bounded functions thus it's easy to
think about two boundary sequences
for this one which approaches the same value.
Let us try to extrapolate
this rule basically for the case of functions.
Let us look for example at
these two equations and these two graph.
Basically, they are looking pretty much the same.
Example, as in sequences because we are
looking at sine divided by
some unbounded function x or in other words we have
some bounded functions sine x and infinitesimal function,
one divided by x which approaches zero.
The idea here drawn in the figure.
This is our, in blue curve,
our function and here comes two basic bound [inaudible].
One divided by x and minus one divided by x.
Since both of them approaches zero,
thus our function approaches zero.
Basically idea here is that if we considering
the project of bounded and infinitesimal function,
the result is infinitesimal.
Since we're actually all right,
zero multiplied by something which has no
definite but has some finite limit but is
just simply bounded is not growing unexpectedly or
just happens to be
some extraordinary large functions from time to time.
Simply bounded multiplied
by infinitesimal, infinitesimal.
Same thing apply for example for
our second function here because what we do
have here is we have x
multiplied by some nightmarish cosine function,
cosine of exponents, of integer part of
one divided by x factorial multiplied by the x.
This is quite, actually frightened.
But since cosine is bounded by a minus one and one,
and the rest of the function is bounded by zero and one,
all these multiply is bounded.
Thus, we actually have
our infinitesimal function and bounded function.
Thus, the limit is simply zero and we can just move on.