- [Instructor] The function h is defined
over the real numbers.
This table gives a few values of h,
so they give us for different x values,
What is the value of h of x?
What is a reasonable estimate
for the limit of h of x as x approaches one?
So with the table we can estimate the limit.
We won't know 100% for sure,
but it's a good way to estimate a limit.
And so what they have here is these first three entries
show what's happening as we approach
x equals one from the left,
from values less than x equals one.
So we can see, well what is h of x for those x values
as we get closer and closer to x equals one?
And we see, let's see what when we're at 0.9,
it's 0.003, it's 3/1000.
Then we get even closer to one,
then the value of our function is negative 4/100.
Then when we get even closer to one,
the value of h is 5/10.
So we have this oscillation in sine
and it seems like the absolute value's
getting larger and larger.
Now if we approach from the right,
we approach x equals one from values larger than one,
so here's we're approaching from the right.
At 1.1 we're at 3/1000.
At 1.01, we are at negative 4/100.
At 1.001 or one and 1/1000,
we are at 5/10.
So this is interesting.
When you have this oscillation of sine,
you say, "Well maybe we might be approaching zero."
You know, maybe something is getting closer and closer
and closer and closer to zero.
Maybe something like that is happening.
But what's troubling here,
the reason why I would say that it's not reasonable,
it's reasonable to think that the limit doesn't exist,
is here as we're getting closer and closer,
here as we're getting closer and closer,
we're getting for the oscillations
are getting more and more significant.
So actually think of it this way.
As we're getting from the left,
as we're getting closer and closer,
the oscillations are getting more significant.
And as we're going from the right
as we get closer and closer,
the oscillations are getting more significant.
Now it looks like there's a little bit of symmetry here.
Let me see if I can draw symmetry reasonably well.
It looks like something like this is happening.
But this is implying as we get,
as we go to 0.9999, then we're going to oscillate again
and we're going to get even further away from zero.
And so we're oscillating around zero,
but where the oscillations are swinging wilder and wilder.
So we're definitely not approaching zero.
We're not getting closer because the absolute value of these
aren't getting closer and closer to zero.
So I would say that the limit doesn't exist.
So a takeaway here is when things start to oscillate,
you're probably already say,
"Okay something fishy is going on."
But that doesn't mean that the limit doesn't always,
it doesn't mean that the limit necessarily doesn't exist
just because of the oscillation.
But if it's oscillating around a value
and as it's getting closer and closer
the oscillations are swinging wilder,
further and further away from that value,
then you have to worry.
And that then the limit probably does not exist.
Let's do another example.
The function h is defined over the real numbers.
This table gives a few values of h,
so it's a similar situation.
What is a reasonable estimate for the limit
as x approaches three of h of x.
So let's see as x approaches three from the left,
as x approaches three from the left,
it looks like, let's see 2.9,
we're at 2/10.
But at 2.99 we go to negative 2/100.
Then a 2.999, we get to positive 2/1000.
So even though it's oscillating,
magnitude-wise, we're getting closer and closer to zero.
So this is an oscillation that you know,
it might look like this.
We're oscillating in sine,
but it looks like we're getting closer and closer
and closer to zero here.
So this looks like we might be,
looks like our function might be approaching zero,
as x approaches three from the left.
Now let's think about from the right.
Well from the right, we see something similar here.
3.1, we're at negative 2/10.
3.01, positive 2/100.
3.001, we're even closer to three, negative 2/1000.
And so it looks like we are definitely oscillating,
but the oscillations are getting less and less wild.
Their absolute values are getting smaller and smaller,
so it actually seems reasonable,
we're oscillating around zero.
And with every swing as we get closer and closer
to x equals three,
the oscillations are getting smaller and smaller.
So once again, we don't know for sure.
We can't prove it mathematically.
But from this table it is reasonable to estimate
that the limit here is zero.
Now these other choices are,
I would call them distractors.
This one down here is interesting
because the function is actually defined
at x equals three and it equals four.
But this might be a situation where the limit is approached,
we're getting closer and closer,
but the function is defined up here at four.
And this reinforces this idea that look,
the limit can approach a value that is different
from the value that the function is defined.
That's actually one of the points of limits,
is to study what is a function approach
even when the function might be defined at a different value
or even when the function isn't defined at that value,
what is it approaching as x approaches that value?
Now as I said, for this situation it's not,
it's reasonable to think that the limit exists.
I don't have any good information to think otherwise.
We don't know 100% sure.
they put the three here,
but we're thinking about what is h approaching
as x approaches three?
Now there's no information here that would imply
that h is approaching three.
And at 0.22,
they're just playing with the digits right over here.
But this is getting further from zero
than what seems like what we're trying to approach,
so I'd rule that one out as well.
But the big takeaway here is
if you start to see, especially on a table or anything
that as you get closer and closer to a value
that the function is oscillating
that should kind of put your spider senses on notice.
But that doesn't necessarily mean
that the limit doesn't exist.
If the swings are getting wilder and wilder
as we approach the value that we're oscillating around,
then the limit probably doesn't exist.
But if the swings, the oscillations,
are getting smaller in magnitude
as we approach the thing that we're oscillating around,
in this case zero,
then is actually is reasonable to think
that the thing that we're oscillating around
actually is the limit.