- [Voiceover] What I would like to tackle in this video
is what I consider to be a particularly
interesting limits problem.
Let's say we want to figure out the limit
as X approaches zero from the positive direction
of sine of X.
This is where it's about to get interesting.
Sine of X to the one over
the natural log of X power
and I encourage you to pause this video
and see if you can have a go at it
fully knowing that this
is a little bit of a tricky exercise.
I'm assuming you have attempted.
Some of you might have been able to figure out
on the first pass.
I will tell you that the first time
that I encountered something like this,
I did not figure it out at the first pass
so definitely do not feel bad
if you fall into that second category.
What many of you all probably did
is you said okay, let me think about it.
Let me just think about the components here.
If I were to think about the limit,
if I were to think about the limit as X approaches zero
from the positive direction of sine of X,
well that's pretty straightforward.
That's going to be zero,
so you could think of like this part of it
is going to approach zero but then if you say,
and you could say, I guess I should say.
The limit as X approaches zero
from the positive direction of one over
natural log of X
and this is why we have to think about it
from the positive direction.
It doesn't make sense to approach it
from the negative direction.
You can't take the natural log
of a negative number.
That's not in the domain for the natural log
but as you get closer and closer to zero
from the negative direction,
the natural log of those values,
you have to raise E
to more larger and larger negative values.
This part over here is going to approach
negative infinity.
It's going to go to negative infinity.
One over negative infinity,
one divided by super large
or large magnitude negative numbers,
well, that's just going to approach zero.
You could say that this right over here
is also going to be,
is also going to be equal to zero.
That doesn't seem to help us much
because if this thing is going to zero
and that thing is going to zero,
it's kind of an implication that well
maybe this whole thing is going to zero
to the zero power but we don't really know what zero,
let me do the some, those color.
Zero to the
zero power but this is one of those great
fun things to think about in mathematics.
There's justifications why this could be zero,
justifications why this could be one.
We don't really know what to make of this.
This isn't really a satisfying answer.
Something at this point
might be going into your brain.
We have this thing that we've been exposed
to called L'Hopital's rule.
If you have not been exposed to it,
I encourage you to watch the video,
the introductory video on L'Hopital's rule.
In L'Hopital's rule, let me just write it down.
L'Hopital's rule
helps us out with situations where
when we try to superficially evaluate the limit,
we get indeterminate forms
things like zero over zero.
We get infinity over infinity.
We get negative infinity over negative infinity
and we go into much more detail into that video.
It seems this is kind of,
it feels like we're getting a zero to a zero.
It's kind of we're getting the strange beast
and at least evokes the notion of L'Hopital's rule.
You will not be, as we'll see in a few seconds,
you're not wrong to, for that
L'Hopital's rule neuron to be triggering in your brain
although you can't apply it directly
to this right over here.
L'Hopital's rule does not apply to
or directly apply to the zero to the zero form
but what we can do is construct a problem
where L'Hopital's rule will apply
and then use that to solve, to figure out
what this is going to be.
This was essentially the tricky part of this exercise.
Well what do I mean?
Well if we set Y
equal to Y of
and maybe let me write it this way.
If we set, and I'll write Y,
I could just write Y but I'll say Y
is clearly a function of X.
If we say Y of X is going to be sine of X,
sine of X to the one over
natural log of X.
This thing right over here is essentially saying
what's the limit as X approaches zero
from the positive direction of Y
and once again we don't know.
Maybe it's zero to the zero
but we don't know what zero to zero actually is.
What we could do, what we could do
and this is a trick that you see a lot
and anytime you get kind of weird things with exponents
and whether you're doing limits or derivatives,
as you'll see it's often times useful
to take the natural log of both sides.
Well what happens if you take the natural log
of both sides here?
On the left-hand side
you're going to have the natural log,
the natural log, and whenever I think of natural log
and E the way I always think about them,
the color green for some bizarre reason
but we'll say the natural log of Y
is equal to.
If you take the natural log of this thing,
actually let me just, I don't want to skip steps here
because this is interesting.
This is going to be the natural log
of all of this business of sine of X,
let me write this way.
Sine of X,
sine, I want to do this in that orange color.
The natural log of
sine of X to the one
over the natural log of X.
Well we know from our exponent prior
our logarithm properties,
the logarithm of something to a power,
that's the same thing as the power.
One over natural log of X
times the logarithm, this case the natural logarithm
of whatever taking the sine of X here.
Sine of X
or we could say the natural log of Y.
Want to keep, stay color consistent
for at least one more step.
The natural log of Y is equal to,
if we just rewrite this
this is going to be the natural log
of sine of X,
the natural log
of sine of X
over the natural log of X.
Well this is all interesting
but why do we care about this?
Why did I do this?
Well instead of thinking about what is the limit?
What is the limit as X approaches zero
from the positive direction of Y?
Let's think about what the natural log of Y
is approaching as we approach,
as X approaches zero from the positive direction.
Let's figure out what the limit of this expression
right over here is as X approaches zero
from the positive direction.
What is a natural log of Y?
What is this whole thing?
Not Y, what is the natural log of Y approaching?
Let's think about that scenario.
Let me write, do this in a new color.
We want to figure out what is the limit
as X approaches zero from the positive direction
of this business and I'll just write it in one color.
The natural log
of sine of X
over the natural log of X.
I don't know, I wrote one time in print,
one time in cursive.
I'll just be consistent right over there.
Now why is this interesting?
Let's see in the numerator here,
this thing's going to approach zero,
natural log of zero you're going to approach
negative infinity.
This thing right over here natural log of,
as you approach it from the positive direction.
Once again you're going to approach negative infinity.
This gives you that indeterminate form.
This is giving you
that indeterminate form of negative infinity
over negative infinity which is neat
because this triggers or at least tells us
that L'Hopital's rule may be appropriate here.
We could say that this is going to be equal to
the limit as X approaches zero
from the positive direction.
Let me give myself a little bit more
real estate to work up with
and I can take the derivative of the numerator
and the derivative of the denominator.
Derivative of the numerator,
so the derivative of the numerator,
I'm going to apply the chain rule here.
Derivative of sine of X is cosine of X
and then the derivative,
the natural log of sine of X with respect
to sine of X is going to be one over sine of X.
This is going to be over sine of X,
so that's the derivative of the numerator.
Then the derivative of the denominator
is just going to be one over X,
one over X.
This is all going to be equal to,
this is equal to the limit
as X approaches zero from the positive direction
of I could write this as cosine of X,
cosine of X.
Let's see,
if I'm dividing by X.
I'm dividing by X, I am going to get,
this is going to be X over sine of X,
X over sine of X.
When I apply
and when I try to take the limit here
I'm going to get a zero,
once again we got a zero over zero.
This doesn't feel too satisfying
but once again this is where our limit properties
might be useful.
As you can tell, this is not the most
trivial of problems
but this is going to be the same thing
and this will take a little bit of pattern recognition.
This is the same thing as
because we know that the limit of the product
of two functions
is equal to the product of their limits,
this is the same thing as the limit,
the limit as X approaches zero
from the positive direction of,
if we take this part,
let me do this in a different color.
If we take this part,
that's not a different color.
If we take this part right over here,
so that's going to be X over sine of X
and then times the limit.
Let me put parenthesis here
times the limit,
the limit
as X approaches zero from the positive direction
of cosine of X,
of cosine of X.
Now this thing right over here
is pretty straightforward.
You can just evaluate it at zero,
you get one.
This thing right over here is equal to one
but what's this thing?
This might ring a bell.
You might have seen the limit as X approaches zero,
of sine of X over X.
This is just the reciprocal of that.
This is X over sine of X
but when you just superficially try to evaluate it.
You get zero over zero so you can then apply
L'Hopital's rule to this thing.
Once again this is quite an interesting scenario
we find ourselves in.
This is going to be the same thing as the limit
as X approaches zero from the positive direction.
Derivative at the top is one,
derivative at the bottom, is cosine of X.
Well, this is just going to be one over
cosine of zero is one,
so this is just going to be equal to one.
We got to apply L'Hopital's rule again
and realize that this limit is going to be equal to one.
One times one is one
so this thing right over here is equal to one.
This thing right over here,
this thing right over here is going to be,
this thing right over here is going to approach one
which tells you that this thing is approaching one.
What do we now know?
We now know and I'll write it out in language.
We now know that the limit
of the natural log of Y,
the limit of the natural log of Y
as X approaches zero from the positive direction
is equal to one.
If the natural log of Y is approaching one,
what is Y approaching?
Well in order for the natural log, so once again
we just know this thing right over here.
We know this thing is one
and this thing is a natural log of Y.
We now know that this thing,
the limit as X approaches zero of this thing
is one and that's the same thing as a limit
as X approaches zero from the positive direction
of the natural log of Y.
These things are equivalent, is equal to one.
Well the natural log of Y is approaching one
so if the natural log of Y
is approaching, let me write it this way.
The natural log of Y is approaching one.
Well what must Y be approaching?
Well to get the natural log of something
that gets you one, well Y must be approaching E
because natural log of E is one.
Then Y must be approaching E
and we are done because that's what we cared about.
We cared about what is Y, this is Y, remember,
we defined this whole thing as Y.
We said what is Y approaching
as X approaches zero from the positive direction?
Well we figured out that the natural log of Y
is approaching one as X approaches zero
from the positive direction.
That means that Y must be approaching E.
This tells us that this thing,
this thing right over here
is equal to E
which is somewhat mind blowing.
We seen other, the E is popping up
and it's involving sine of X
and of course natural log of X,
you expect E to be involved somehow.
It is a pretty fascinating problem in my mind.