www.mathcentre.ac.uk/.../Limits%20of%20functions.mp4

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.