all right in this video I just want to
do some problems some basic problems
related to limits and graphs basically
being able to look at a graph and
evaluate a limit I think it's you know
limits are something you do a lot in
calculus and I think to do a lot of
limits I think about graphs of functions
so it's kind of a fundamental idea so
nothing hopefully too heavy this is
stuff that you know certainly can be
confusing the first time or the second
time or the third time you see it but
this is the basic idea so if you look at
our graph here you know y equals f of X
notice it's a there's a little break in
it at the x coordinate of 3 there's a
little hole so there's a point of
discontinuity and that's fine
nothing nothing going on there just you
know one little observation here so just
to pick on a point in particular notice
if you take an x coordinate that gets
closer and closer say two x equals three
if we take an x value that gets closer
and closer to three whether we kind of
come from the left side of three that
means using smaller values obviously or
from the right side of three and that
means we're using well values larger
than three if you think about on the
corresponding graph what are the Y
values getting closer to four those X
values well to me it looks like as X
gets closer and closer and closer to
three the Y values on the graph are
simply getting closer and closer to two
so no big deal so I went ahead and wrote
that down basically it says again as X
gets closer to three f of X or
equivalently the Y values those get
closer to the number two
notice it's even undefined too because
of the open circle still you know as X
gets closer to 3 the Y values
numerically get closer to 2 we write
that as as the following we say the
limit as X approaches 3 of the function
f of X equals 2 so to me when I see this
notation all it says is it says you've
got some graph y equals f of X if you
plug in stuff close to 3 it says hey the
Y values get close to 2 we write an
equal sign I think that's a little
misleading because it never actually
equals 2 but it gets as close to it as
you want and that's the idea so let's
look at a couple problems here with a
graph and we'll just try to answer some
of these ok so notice my little graph
here it's got to break it negative 2
it's got a vertical asymptote at x
equals 2 there's another little break
here in x equals 4 so the first question
here just says what is f of 0 well all
that means is right you're just plugging
in x equals 0 what's the corresponding Y
value well in this case F of 0 would
just equal the Y value of 2 hopefully
you can see that likewise this limit
question it says okay what's the limit
as X approaches 0 of f of X so again
that to me says as X gets closer to 0
what do the Y values get closer to well
here's X equals zero so I just find the
corresponding graph and again I think
well as X gets closer and closer to zero
on the graph the Y values are getting
closer and closer to the number two no
matter which side you come from so in
this case we'll say the limit as X
approaches zero of f of X is also two
and this is something that's important
you'll see this in another part of the
problem here this has to be a unique
number if not we simply say the limit
doesn't exist okay so what if we ask
what's F of negative 2
well that simply says if you plug
negative 2 into the function what do you
get out what's the corresponding y-value
well this simply says if you plug
negative 2 n since our dot is up here at
5 it says if you plug negative 2 n well
you get 5 out okay however if we look at
the limit as X approaches negative 2
again that simply means if we take
x-coordinates close to negative 2 if we
take x-coordinates close to negative 2
what are the corresponding y-values
getting closer and closer to and in this
case it looks like if X gets closer and
closer to negative 2 to me the Y values
are getting closer and closer to 1 so in
this case I would say the limit as X
approaches negative 2 is just 1 so again
this is important it doesn't actually
matter what happens at negative 2 it
just happened the only thing that's
important is what happens when you get
close to negative 2 I get close to
negative 2 the Y values get close to 1
ok so let's see a couple more here
suppose we do the limit as X approaches
positive 2 of our function f of X well
in this case as X approaches 2 the Y
values on the graph well they just get
bigger and bigger and bigger and bigger
and it doesn't matter whether you come
for the left or the right they just get
larger larger and larger sometimes if
they go to infinity people will say it
does not exist in this case I would
actually write that I would say well the
limit goes to positive infinity and what
that means is again it just says the
graph is spiking off ok let's see just a
few more here F of 4 that says if you
plug 4 in what Y value do you get out
well if you plug 4 in it looks like
you're getting out the Y value of 1
and just a couple more questions here
let's do suppose we ask ourself what's
the limit as X approaches four I left
this one out well notice in this case it
kind of depends because if you take you
know x coordinates that are close to
four but smaller than four
well those Y those Y values get closer
and closer to one right so X values to
the left of four that means I'm going to
be on the graph just a little bit to the
left of 4 and the Y values get closer to
1 well but if you do it you know for X
values just a little bit bigger than 4
well now you're on the upper portion of
the graph and then it says the Y values
get closer and closer to I don't know
we'll pretend it's lining up here at
positive 4 so you can't say two numbers
you can't say it's like 1 and 4 you know
in this case if the limits aren't the
same from opposite sides and we have a
better way of saying this we simply say
that the limit does not exist in that
case so to me basically it says if
there's a break or a jump in the graph
the limit doesn't exist
however we can talk about limits from
the left and limits from the right on
graphs so this will be the last two
problems here what we'll do is we'll
write the limit as X approaches 4 and
that we've got a little positive sign
what that means is from the right of the
number 4 and if we're to the right of
the number 4 that means we're using
numbers well that are larger than 4 so
in this case if we take x coordinates
that are larger than 4 so bigger than 4
but we're getting closer to 4 if you
think about the corresponding Y values
on the graph well those Y values are
getting closer and closer to the number
4 so we would say the limit as X
approaches 4 from the right is 4 and if
we do the limit as X approaches 4 from
the left ok so if we take numbers a
little bit smaller than 4 well now the Y
values on the graph are getting closer
and closer to the number positive 1 so
we would say the limit as X approaches 4
from the left of f of X is positive 1 so
a lot of a lot of stuff kind of crammed
into this video over limits so again if
it's the first time you're seeing limits
and graphs I think they're definitely
you know can be a little confusing but
it's definitely an idea that you've got
to master before you can really I think
if not you know things just you know
limits don't go away and if you don't
get a handle on them they get confusing
you know everything stays confusing and
probably again I think the best way to
think about limits a lot of times to
visualize is to really understand them
graphically so all right I hope this
video makes some sense and helps you out
out there