In this video, we'll define what it means for a function F to
tend to a limit as X tends to
Infinity. We also define what it means for its tend to limit as X
tends to minus Infinity, and as X tends to a real number.
Remember from the video limits of sequences, we define what it
meant for a sequence to tend to A to a limit as N tends to
Infinity, we said. Sequence wired tends to limit L
as an tends to Infinity.
If however, small a number I
chose. The sequence YN would get that close to L and stay that
close. One of our examples was the sequence YN equals
one over N.
I just put a graph of this.
So her plot the points of why and which was
one equals one over N.
It looks like this sort of it 1.
And then two was a half. Then it
went 1/3. Now this sequence tended to
0 because eventually.
The points in the sequence get
as close. Zeros alike, so have a smaller distance I choose.
The points in the sequence will eventually get within
that boundary from zero and stay in there.
The functions are defined in a similar way to limits
of sequences.
If we have a function like FX equals one over X.
Where I've actually already lost some of the points for
this already, so I'll use it if we sketch the graph.
You can see that this function gets closer and closer to 0.
In fact, whichever tiny distance
we choose. Like this distance I've already sketched in here.
Whatever distance we choose this function will get within that
from zero and stay that close.
If a function does this, we say it tends to limit
O's extends to Infinity.
If a function tends to zeros extends to Infinity, we write it
like this. Who writes FX tends to 0?
As X tends to Infinity.
For sometimes the limits of FX as X tends to Infinity equals 0.
Here's another function with
the limit.
This time will have F of X.
Equals 3 - 1 over X squared, 4X greater than 0.
So I'll just sketch this graph.
Will have three about here.
Right?
This graph looks something like this.
Now again, whatever tiny distance I choose around 3:00.
It doesn't matter how small this is.
This function.
Eventually gets trapped that far away from 3.
So this function has limit three, and again we write FX
tends to three.
As X tends to Infinity.
For the limit of FX.
Equals 3 as X tends to Infinity.
In general, we say that the function has limit L as X tends
to Infinity. If how the smaller distance he choose.
The function gets closer than that and stays closer than that
as X gets larger.
Not all functions have real limits as X tends
to Infinity. Here's one that doesn't.
Will have F
of X. Equals X squared.
And that is, I'm sure, you know, looks something like
this.
Now as X gets larger, this certainly doesn't get any closer
to any real number.
In fact, however, large number I choose.
This function will eventually get bigger and stay bigger than
it as X goes off to Infinity.
If a function does this, we say that it has limits Infinity as X
tends to Infinity. We write F of X tends to Infinity.
As X tends to Infinity.
Or limits. With X as X tends to Infinity equals
Infinity. If we have a function like
F of X equals
minus X. It looks like this.
This also doesn't tend to a real limit as X tends to
Infinity. But it doesn't tend to Infinity either, because it
doesn't get very large.
In fact, have a large and
negative. A number I choose.
This function will get below that number and stay below it.
If a function does this, we'd say it tends to minus Infinity,
as X tends to Infinity.
So we write. F of X.
Tends to minus Infinity.
As X tends to Infinity.
Oregon limit of F of X.
Equals minus Infinity as X tends to Infinity.
Some functions that have any limits at all,
as X tends to Infinity.
This is the graph FX equals X sign X.
But as you can see, it certainly doesn't get closer to any real
number. As extends to Infinity.
Also, if I pick a really large number.
The graph will eventually get
above it. But it won't stay above it because it always
come back down to zero and go negative.
Also, for pick a really large negative number, it will go
below it, but it won't stay below it because it'll go
back up to zero and then go positive again.
So this function doesn't tend to Infinity, and it doesn't tend to
in minus Infinity as well.
So this function doesn't have any limited tool.
We can also define limits for functions as X tends to
minus Infinity.
Here's a graph of E to the X.
So. This after 0 gets
large. The 40
Gets closer and closer to 0.
As X gets more negative here.
This function gets closer and closer to 0.
And however small a distance I choose.
The function will get closer than that to zero and stay
closer than that as X gets more
negative. If a function is this
we say? F of X tends
to. 0. As X tends to minus Infinity.
Oh, limit as X tends to minus Infinity of F of X.
Equals 0.
In general, we say a function has limit L's extends to minus
Infinity. If how the smaller distance I choose the function
gets closer than that and stays closer than that as X gets more
negative. If we have a graph
like X squared.
Shoulder sketch.
You can see that this doesn't get closer to any real number as
X goes to minus Infinity.
But however larger number I choose.
As X gets more negative.
This graph will go above that number and stay above
that number.
If a function does this, we say it tends to Infinity as X tends
to minus Infinity. So F of X tends to Infinity.
As X tends to minus Infinity.
Limits of F of X equals Infinity as X tends to minus Infinity.
Now function like X cubed doesn't do either
of these things. The graph is something like
this.
So it certainly doesn't get larger than any number I pick as
X goes to minus Infinity and stay larger than it.
And it doesn't get closer to any real number, so it doesn't have
a real limit. But however larger negative number I choose.
As X gets more negative, this function will drop below that
number and stay below it and this works for any number I can
choose. When this happens, we say F of X tends to minus
Infinity. As X tends to minus
Infinity. For, again, the limits of F of X.
Equals minus Infinity.
As X tends to minus Infinity.
Now again, a function didn't have any limit as X tends to
minus Infinity. And our graph of FX equals X sign X is a
good example of this again.
As X gets more negative, this certainly doesn't get closer to
any real number, so it doesn't have a real limit.
It might go above any number we can pick, but it won't stay
above it because it comes back down and goes through zero.
And it won't get more. It won't get more and stay more negative
than any large negative number we can choose, because always
come back up to zero and go
positive again. So this is a function that has no limit as X
tends to minus Infinity.
There's one more type of limit. We can
define functions. Let's look at the graph of the
really simple function like F of X equals X +3.
So that looks something like
this. Three there.
If I pick a point on this graph,
like say.
Here, when X is one.
The graph goes through 1 four.
You can see that as X approaches
one. The value of the function approaches 4.
We say that.
F of X the limit of F of X.
Equals 4 as X tends to one.
Similarly.
If we say X equals 5.
Then the function.
Is it 8? And his ex gets closer to five. The function gets close
to 8. So the limit of F of X as X tends to 5
equals 8.
Now this doesn't look very
useful. But it does become useful if we have functions that
aren't defined data point.
This is the case with this function. This is F
of X. Equals.
X sign one over X.
You can see for this function that as X gets close to 0.
The function gets very very
close to 0. But the cause at X equals 0. We have a zero
denominator. This function isn't defined for X equals 0.
But because the function gets closer and closer to zero, it's
almost as if the value of the function at zero should be 0.
So the limit of F of X.
As X tends to 0 equals 0.
And you can think of 0.
As the value F of X should take when X equals 0, even
though it's not defined.
Here's another example.
This function is F of X equals Y
to the minus one over X squared.
Now again, when X equals 0, we have a zero denominator. Here in
the function. So it's not defined at X equals 0.
But as X gets close to 0.
This function approaches 0, so again the limit of F of X as X
tends to zero in this case is zero, and again we can think of
0. Is the value F should take when X equals 0.
Not all functions have nice limits
like this.
If we look at the function MoD X over XF of X
equals MoD X over X.
If X is positive.
Then The top of this
is just X.
So this becomes X over X, which is one.
But if X is less than 0.
This becomes little becomes minus X, so that's minus X over
X. So that's minus one.
When X is 0, is not defined at all, so the graph looks like
this. It's just one.
When X is greater than zero and
it's just. Minus one.
When X is less than 0.
And you can see here there's no way we can define a good limit
for this, because as we approach zero from the right hand side.
It looks like it should be one, but if we approach zero from the
left hand side, the function looks like it should be taking
value minus one. So here it's not clear what ever vex should
be when X is 0.
Because there isn't a proper limit for this function as X
tends to 0, so this is a good example of a function that
doesn't have a limit as X tends to a number.