WEBVTT

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In this video, we'll define what
it means for a function F to

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tend to a limit as X tends to

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Infinity. We also define what it
means for its tend to limit as X

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tends to minus Infinity, and as
X tends to a real number.

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Remember from the video limits
of sequences, we define what it

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meant for a sequence to tend to
A to a limit as N tends to

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Infinity, we said. Sequence
wired tends to limit L

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as an tends to Infinity.

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If however, small a number I

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chose. The sequence YN would get
that close to L and stay that

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close. One of our examples
was the sequence YN equals

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one over N.

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I just put a graph of this.

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So her plot the points
of why and which was

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one equals one over N.

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It looks like this sort of it 1.

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And then two was a half. Then it

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went 1/3. Now this
sequence tended to

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0 because eventually.

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The points in the sequence get

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as close. Zeros alike, so have a
smaller distance I choose.

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The points in the sequence
will eventually get within

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that boundary from zero and
stay in there.

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The functions are defined
in a similar way to limits

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of sequences.

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If we have a function like FX
equals one over X.

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Where I've actually already
lost some of the points for

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this already, so I'll use
it if we sketch the graph.

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You can see that this function
gets closer and closer to 0.

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In fact, whichever tiny distance

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we choose. Like this distance
I've already sketched in here.

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Whatever distance we choose this
function will get within that

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from zero and stay that close.

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If a function does this,
we say it tends to limit

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O's extends to Infinity.

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If a function tends to zeros
extends to Infinity, we write it

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like this. Who writes FX
tends to 0?

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As X tends to Infinity.

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For sometimes the limits of FX
as X tends to Infinity equals 0.

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Here's another
function with

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the limit.

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This time will have F of X.

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Equals 3 - 1 over X squared,
4X greater than 0.

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So I'll just sketch this graph.

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Will have three about here.

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Right?

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This graph looks
something like this.

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Now again, whatever tiny
distance I choose around 3:00.

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It doesn't matter
how small this is.

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This function.

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Eventually gets trapped that far
away from 3.

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So this function has limit
three, and again we write FX

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tends to three.

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As X tends to Infinity.

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For the limit of FX.

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Equals 3 as X tends to Infinity.

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In general, we say that the
function has limit L as X tends

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to Infinity. If how the smaller
distance he choose.

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The function gets closer than
that and stays closer than that

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as X gets larger.

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Not all functions have
real limits as X tends

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to Infinity. Here's one
that doesn't.

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Will have F

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of X. Equals X squared.

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And that is, I'm sure, you
know, looks something like

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this.

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Now as X gets larger, this
certainly doesn't get any closer

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to any real number.

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In fact, however, large
number I choose.

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This function will eventually
get bigger and stay bigger than

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it as X goes off to Infinity.

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If a function does this, we say
that it has limits Infinity as X

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tends to Infinity. We write F
of X tends to Infinity.

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As X tends to Infinity.

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Or limits. With X as
X tends to Infinity equals

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Infinity. If we
have a function like

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F of X equals

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minus X. It
looks like this.

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This also doesn't tend to a
real limit as X tends to

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Infinity. But it doesn't tend to
Infinity either, because it

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doesn't get very large.

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In fact, have a large and

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negative. A number I choose.

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This function will get below
that number and stay below it.

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If a function does this, we'd
say it tends to minus Infinity,

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as X tends to Infinity.

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So we write. F of X.

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Tends to minus Infinity.

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As X tends to Infinity.

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Oregon limit of F of X.

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Equals minus Infinity
as X tends to Infinity.

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Some functions that
have any limits at all,

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as X tends to Infinity.

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This is the graph FX
equals X sign X.

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But as you can see, it certainly
doesn't get closer to any real

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number. As extends to Infinity.

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Also, if I pick a really
large number.

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The graph will eventually get

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above it. But it won't stay
above it because it always

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come back down to zero and
go negative.

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Also, for pick a really large
negative number, it will go

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below it, but it won't stay
below it because it'll go

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back up to zero and then go
positive again.

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So this function doesn't tend to
Infinity, and it doesn't tend to

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in minus Infinity as well.

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So this function doesn't
have any limited tool.

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We can also define limits
for functions as X tends to

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minus Infinity.

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Here's a graph of E to the X.

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So. This after 0 gets

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large. The 40

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Gets closer and closer to 0.

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As X gets more negative here.

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This function gets closer and
closer to 0.

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And however small a
distance I choose.

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The function will get closer
than that to zero and stay

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closer than that as X gets more

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negative. If a function is this

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we say? F of X tends

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to. 0. As
X tends to minus Infinity.

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Oh, limit as X tends to minus
Infinity of F of X.

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Equals 0.

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In general, we say a function
has limit L's extends to minus

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Infinity. If how the smaller
distance I choose the function

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gets closer than that and stays
closer than that as X gets more

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negative. If we
have a graph

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like X squared.

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Shoulder sketch.

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You can see that this doesn't
get closer to any real number as

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X goes to minus Infinity.

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But however larger
number I choose.

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As X gets more negative.

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This graph will go above
that number and stay above

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that number.

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If a function does this, we say
it tends to Infinity as X tends

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to minus Infinity. So F of
X tends to Infinity.

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As X tends to minus Infinity.

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Limits of F of X equals Infinity
as X tends to minus Infinity.

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Now function like X
cubed doesn't do either

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of these things. The
graph is something like

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this.

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So it certainly doesn't get
larger than any number I pick as

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X goes to minus Infinity and
stay larger than it.

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And it doesn't get closer to any
real number, so it doesn't have

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a real limit. But however larger
negative number I choose.

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As X gets more negative, this
function will drop below that

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number and stay below it and
this works for any number I can

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choose. When this happens, we
say F of X tends to minus

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Infinity. As X tends to minus

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Infinity. For, again, the limits
of F of X.

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Equals minus Infinity.

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As X tends to minus Infinity.

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Now again, a function didn't
have any limit as X tends to

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minus Infinity. And our graph
of FX equals X sign X is a

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good example of this again.

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As X gets more negative, this
certainly doesn't get closer to

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any real number, so it doesn't
have a real limit.

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It might go above any number we
can pick, but it won't stay

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above it because it comes back
down and goes through zero.

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And it won't get more. It won't
get more and stay more negative

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than any large negative number
we can choose, because always

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come back up to zero and go

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positive again. So this is a
function that has no limit as X

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tends to minus Infinity.

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There's one more type
of limit. We can

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define functions. Let's
look at the graph of the

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really simple function
like F of X equals X +3.

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So that looks something like

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this. Three there.

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If I pick a
point on this graph,

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like say.

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Here, when X is one.

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The graph goes through 1 four.

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You can see that as X approaches

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one. The value of the
function approaches 4.

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We say that.

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F of X the limit of F of X.

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Equals 4 as X tends to one.

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Similarly.

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If we say X equals 5.

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Then the function.

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Is it 8? And his ex gets closer
to five. The function gets close

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to 8. So the limit of F
of X as X tends to 5

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equals 8.

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Now this doesn't look very

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useful. But it does become
useful if we have functions that

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aren't defined data point.

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This is the case with
this function. This is F

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of X. Equals.

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X sign one over X.

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You can see for this function
that as X gets close to 0.

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The function gets very very

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close to 0. But the cause at X
equals 0. We have a zero

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denominator. This function isn't
defined for X equals 0.

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But because the function gets
closer and closer to zero, it's

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almost as if the value of the
function at zero should be 0.

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So the limit of F of X.

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As X tends to 0 equals 0.

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And you can think of 0.

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As the value F of X should
take when X equals 0, even

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though it's not defined.

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Here's another example.

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This function is F
of X equals Y

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to the minus one
over X squared.

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Now again, when X equals 0, we
have a zero denominator. Here in

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the function. So it's not
defined at X equals 0.

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But as X gets close to 0.

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This function approaches 0, so
again the limit of F of X as X

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tends to zero in this case is
zero, and again we can think of

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0. Is the value F should take
when X equals 0.

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Not all functions
have nice limits

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like this.

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If we look at the function
MoD X over XF of X

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equals MoD X over X.

00:15:36.510 --> 00:15:38.070
If X is positive.

00:15:39.020 --> 00:15:44.578
Then The
top of this

00:15:44.578 --> 00:15:46.879
is just X.

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So this becomes X over X,
which is one.

00:15:52.960 --> 00:15:55.214
But if X is less than 0.

00:15:55.840 --> 00:16:01.043
This becomes little becomes
minus X, so that's minus X over

00:16:01.043 --> 00:16:03.510
X. So that's minus one.

00:16:04.550 --> 00:16:08.694
When X is 0, is not defined at
all, so the graph looks like

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this. It's just one.

00:16:12.480 --> 00:16:14.566
When X is greater than zero and

00:16:14.566 --> 00:16:16.900
it's just. Minus one.

00:16:19.650 --> 00:16:21.006
When X is less than 0.

00:16:22.590 --> 00:16:26.300
And you can see here there's no
way we can define a good limit

00:16:26.300 --> 00:16:29.480
for this, because as we approach
zero from the right hand side.

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It looks like it should be one,
but if we approach zero from the

00:16:33.880 --> 00:16:36.850
left hand side, the function
looks like it should be taking

00:16:36.850 --> 00:16:41.252
value minus one. So here it's
not clear what ever vex should

00:16:41.252 --> 00:16:42.842
be when X is 0.

00:16:43.390 --> 00:16:46.349
Because there isn't a proper
limit for this function as X

00:16:46.349 --> 00:16:49.846
tends to 0, so this is a good
example of a function that

00:16:49.846 --> 00:16:52.536
doesn't have a limit as X
tends to a number.