right over here and we want to figure out

three different limits and like always

pause this video and see if you can figure it out

on your own before we do it together.

Alright now first let's think about what's the limit

of f of x it's x approaches six.

So as x, I'm gonna do this in a color you can see,

as x approaches six from both sides

well as we approach six from the left hand side,

from values less than six,

it looks like our f of x is approaching one

and as we approach x equals six from the right hand side

it looks like our f of x is once again approaching one

and in order for this limit to exist,

we need to be approaching the same value

from both the left and the right hand side

and so here at least graphically,

so you never are sure with a graph but this is

a pretty good estimate, it looks like we are approaching one

right over there, in a darker color.

Now let's do this next one.

The limit of f of x is x approaches four

so as we approach four from the left hand side

what is going on?

Well as we approach four from the left hand side

it looks like our function, the value of our function

it looks like it is approaching three.

Remember you can have a limit exist at an x value

where the function itself is not defined,

the function , if you said after four, it's not defined

but it looks like when we approach it from the left

when we approach x equals four from the left

it looks like f is approaching three

and then we approach four from the right,

once again, it looks like our function is approaching three

so here I would say, at least from what we can tell

from the graph it looks like the limit

of f of x is x approaches four is three,

even though the function itself is not defined yet.

Now let's think about the limit as x approaches two.

So this is interesting the function is defined there

f of two is two, let's see when we approach

from the left hand side it looks like our function

is approaching the value of two

but when we approach from the right hand side,

when we approach x equals two from the right hand side,

our function is getting closer and closer to five

it's not quite getting to five but as we go from

you know 2.1 2.01 2.001 it looks like our function

the value of our function's getting closer and closer

to five and since we are approaching two different values

from the left hand side and the right hand side

as x approached two from the left hand side

and the right hand side we would say that this limit

does not exist so does not exist.

Which is interesting.

In this first case the function is defined at six

and the limit is equal to the value of the function

at x equals six, here the function was not defined

at x equals four, but the limit does exist

here the function is defined at f equals, x equals two

but the limit does not exist as we approach x equals two

let's do another function just to get more cases

of looking at graphical limits.

So here we have the graph of Y is equal to g of x

and once again pause this video and have a go at it

and see if you can figure out these limits graphically.

So first we have the limit as x approaches five

g of x so as we approach five from the left hand side

it looks like we are approaching this value

let me just draw a straight Line that takes us

so it looks like we're approaching this value

and as we approach five from the right hand side

it also looks like we are approaching that same value.

And so this value, just eye balling it off of here

looks like it's about .4 so I'll say this limit

definitely exists just when looking at a graph

it's not that precise

so I would say it's approximately 0.4

it might be 0.41 it might be 0.41456789

we don't know exactly just looking at this graph

but it looks like a value roughly around there.

Now let's think about the limit of g of x

as x approaches seven so let's do the same exercise.

What happens as we approach from the left

from values less than seven 6.9, 6.99, 6.999

well it looks like the value of our function

is approaching two, it doesn't matter

that the actual function is defined g of seven is five

but as we approach from the left,

as x goes 6.9, 6.99 and so on,

it looks like our value of our function

is approaching two, and as we approach x equals seven

from the right hand side it seems like the same thing

is happening it seems like we are approaching two

and so I would say that this is going to be equal to two

and so once again, the function is defined there

and the limit exists there but the g of seven

is different than the value if the limit of g of x

as x approaches seven.

Now let's do one more.

What's the limit as x approaches one.

Well we'll do the same thing,

from the left hand side, it looks like we're going

unbounded as x goes .9, 0.99, 0.999 and 0.9999

it looks like we're just going unbounded towards infinity

and as we approach from the right hand side

it looks like the same thing is happening

we're going unbounded to infinity.

So formally, sometimes informally people will say

oh it's approaching infinity or something like that

but if we wanna be formal about what a limit means

in this context because it is unbounded

we would say that it does not exist.

Does not exist.