Constructing multiple functions with same limit

- [Instructor] So we have the graph

of y is equal to g of x right over here.

And I wanna think about what is the limit

as x approaches five

of g of x?

Well we've done this multiple times.

Let's think about what g of x approaches

as x approaches five from the left.

g of x is approaching negative six.

As x approaches five from the right,

g of x looks like it's
approaching negative six.

So a reasonable estimate
based on looking at this graph

is that as x approaches five,

g of x is approaching negative six.

And it's worth noting

that that's not what g of five is.

g of five is a different value.

But the whole point of this video

is to appreciate all that a limit does.

A limit only describes
the behavior of a function

as it approaches a point.

It doesn't tell us exactly
what's happening at that point,

what g of five is,

and it doesn't tell us much

about the rest of the function,

about the rest of the graph.

For example, I could construct
many different functions

for which the limit as x approaches five

is equal to negative six,

and they would look very
different from g of x.

For example, I could say the limit of

f of x

as x approaches five

is equal to negative six,

and I can construct an
f of x that does this

that looks very different than g of x.

In fact if you're up for it,

pause this video and see
if you can so the same,

if you have some graph paper,
or even just sketch it.

Well the key thing is that
the behavior of the function

as x approaches five from both sides,

from the left and the right,

it has to be approaching negative six.

So for example,

a function that looks like this,

so let me draw f of x,

an f of x that looks like this,

and is even defined right over there,

and then does something like this.

That would work.

As we approach from the left,

we're approaching negative six.

As we approach from the right,

we approaching negative six.

You could have a function

like this, let's say the limit,

let's call it h of x,

as x approaches five

is equal to negative six.

You could have a function like this,

maybe it's defined up to there,

then it's you have a circle there,

and then it keeps going.

Maybe it's not defined at
all for any of these values,

and then maybe down here

it is defined for all x values
greater than or equal to four

and it just goes right
through negative six.

So notice,

all of these,

all of these functions
as x approaches five,

they all have the limit defined

and it's equal to negative six,

but these functions all look
very very very different.

Now another thing to appreciate is

for a given function,

and let me delete these.

Oftentimes we're asked to find the limits

as x approaches some type
of an interesting value.

So for example,

x approaches five,

five is interesting right over here

because we have this point discontinuity.

But you could take the limit

on an infinite number of points

for this function right over here.

You could say the limit

of g of x

as x approaches,

not x equals, as x approaches,

one, what would that be?

Pause the video and try to figure it out.

Well let's see,

as x approaches one
from the left-hand side,

it looks like we are
approaching this value here.

And as x approaches one
from the right-hand side,

it looks like we are
approaching that value there.

So that would be equal to g of one.

That is equal to g of one

based on that would be a reasonable,

that's a reasonable conclusion to make

looking at this graph.

And if we were to
estimate that g of one is,

looks like it's approximately negative 5.1

or 5.2, negative 5.1.

We could find the limit

of g of x

as x approaches pi.

So pi is right around there.

As x approaches pi from the left,

we're approaching that value

which just looks actually pretty close

to the one we just thought about.

As we approach from the right,

we're approaching that value.

And once again in this case,

this is gonna be equal to g of pi.

We don't have any interesting
discontinuities there

or anything like that.

So there's two big takeaways here.

You can construct many different functions

that would have the same limit at a point,

and for a given function,

you can take the limit
at many different points,

in fact an infinite number
of different points.

And it's important to point that out,

no pun intended,

because oftentimes we
get used to seeing limits

only at points where something

strange seems to be happening.