Since we already have a pretty good idea about what limits are in an intuitive
sense, we're gonna now take a look at the Heaviside function and in doing so we're
gonna take another look at limits in detail (particularly left sided and right
sided limits). So the Heaviside function looks like this right over here. Now
what's interesting about this function is that when the X, or in this case we
call it 'n,' when the 'n' is less than 0 then the Y value will be 0. But as soon as the
'n' is equal to 0 or greater which, we can see from the graph right here, then the
value for Y will be equal to 1. So a good example of a Heaviside function would be
a switch where 0 is the point and X of 0 would be the point where it would maybe
be representing a light that goes from off to on, 0 to 1 so-to-speak. Now as we
look at the Heaviside function let's consider what the limit would be as it
approaches (as X approaches) 0, particularly starting from the left side.
So we don't have a table, but we know for a fact that all values going towards 0
right over here (all values going towards 0) will have the value of 0 for Y and so
we know that it seems like for now that the limit as X approaches 0 will be
equal to 0 but the problem here now is that when you look at it from the right
side, all values of going towards an X of 0 this way from the positive direction
seems to be yielding a value of one for 'y.' That would make it seem as if the
limit is 1 as opposed to 0. So the question is what do we say when somebody
says find me the limit for this function as X approaches zero? Well the answer to
that would be to just write: "Does Not Exist" and the reason why is that when we talk
about X approaching 0 this is something we went through in the last video but
we're gonna go through in detail this time what we're actually saying is "as X
approaches 0 specifically from both the left and the right side." Now from the
left side, it seemed to be approaching 0 and from the right side it seemed to be
approaching 1 and because it seems to be approaching two different numbers you
can see that again the X's going this way and the X's going this way when
it's going from the left to the right towards 0 it seems to be approaching 0
when it's going from the right to the left approaching 0 for X it seems to be
approaching 1 so when it's approaching 2 different numbers based on if you're
starting from the left side looking towards the right or if you're starting
from the right side looking towards the left then you can say for a fact that
this limit in general "does not exist." However, you cannot say that the limit
does not exist as it pertains to only the left side or only the right side. So
what do I mean by that well what I mean by that is this: you can actually write
if you remember the limit notation you can actually write the entire limit
notation so long as you write this small negative sign right over here to denote
that the limit from the left (so the limit from all the X values starting
from lower than 0 - graphically it would be all
X values starting from lower than zero - basically only in this one direction the
left-sided limit) is equal to zero. You can also refer to the right-sided limit
so the limits starting from all the X values above zero so basically only the
ones looking from this direction towards zero. You can actually say that the limit
for that would be equal to 1. So basically what we just learned is in the
first video we talked about X approaching 0 overall in general which
implies both left and right now we learned that you can actually break them
into specific pieces for example the limit as X approaches 0 from the left or
the limit as X approaches 0 from the right. Awesome! And, in detail, if you can
summarize what we just said, if the limit from the left does not equal the limit
from the right then we can say that the limit overall does not exist. Now if the
limit from the left did equal the limit from the right then you can say that the
limit does exist. More specifically, if let's say the number that the limit
equals to ... I'm going to use a variable 'b' ... then if the limit on the Left equals 'b'
and the limit on the right equals the same number 'b' then we can say not only
that the limit overall exists but we can even specify that the limit overall is
equal to 'b'.
Now that we learned a little bit about one-sided limits limits from the left
from the right and what it would mean when limits from the left and the right
either equal each other or do not equal each other, let's try a few questions
together. So example 1 says, find the limit as X approaches negative 2
now to answer this question why don't we look at all of the options. So it says in
option A that the limit is negative 2. It seems like maybe this answer seems to be
alluding to this negative 2 on the x-axis, which has nothing to do with the
Y value that is being approached. Remember, the limit looks at the Y value
that's being approached so ... I mean ... this -2 has nothing to do with us
definitely (a) is not the answer. (b) says 2 ... I mean ... if we were looking at the
right-sided limit (every x value going this way as X approaches negative 2 from
the right) then (b) could have been the answer but we're certainly not looking
only at the right side, so (b) is not the answer. (c) says 4 so this could be the
answer if we were looking from the x-values on the left side but again the
question never said left side or right side which automatically implies both
left and right so the only option left is the fact that the limit does not
exist and this is definitely the case. In fact, if we want to go into detail we can
say that the limit as X approaches negative 2 from the left for this
function, which I will just call f(x) for now (in fact why don't we just label
it f(x) for this entire function) is equal to four. Now we could also say
that the limit as X approaches negative 2 from the right for this function is
equal to 2. And like we said earlier when we're approaching the same number but
from different directions and the limit does not equal each other then
definitely that limit overall does not exist. If they do equal each other then
whatever number we find over here would be our answer if somebody asked us what
is the limit as X approaches negative 2 overall. In this case once again, since
the left-sided limit does not equal the right-sided limit, we cannot say that
there actually is a limit as X approaches negative 2 overall. So 'd' is
are answer. So in example 2, we have: "Based on the following clues, does the limit
exist for the function f(x) as x approaches 4? So there are two clues and
with this you should be able to say if the limit exists or does not exist. What
do you think? Alright so for a limit to exist as we said earlier if the
left-sided limit is equal to the right-sided limit then we can say that
the limit exists. if we look at the first point over here what it's saying is that
for the function f(x), the limit ... the y value that's being approached as X
approaches 4 (and notice that negative sign) from the left side is equal to 5
you look at the second point here we basically set the same thing only we're
saying "from the right side" - you can see that positive sign. So the limit as X
approaches 4 from the right side for the function f(x) is also equal to 5 and
what do you know the left side and the right side equals each other so what can
we say in general? We can say that the limit as X approaches 4 for the function
f of X and notice how we don't have a negative or positive sign here
what that means is once again that both sides equal to five. So not only can we say
that the limit does exist, we could have also even said, more specifically, that the
limit exists and the limit is equal to five. So this is our answer. Now let's try
our final question together. It says based on the following graph does the
limit exist for the function G(x) as X approaches negative two point five.
What would you say? Well we can see that we're looking at an X approaching
negative two point five. There are no directions attached to this phrase so
we're not talking about left side only or right side only. We're definitely
talking about both sides. When we consider X approaching -2.5
from the left side, you can see that we are getting closer and
closer to (0and unfortunately this is not labeled but it doesn't need to be
labeled because you can see that it's getting closer and closer to the same
number on the right side for the right sided limit. So the left-sided limit is
going to be equal to the right-sided limit because you can see that they are
approaching the same unfilled dot. Whether it was filled ... it could have been filled,
they would have still been approaching the same limit but in this case it's not
filled and it didn't need to be labeled maybe that was I don't know let's say 2?
Or maybe the graph could have said 'a' just to keep it as a variable, but you
would not actually need to even know what the y-value is because you know one
thing for a fact from the left as it pertains to an X of 2.5 it is
approaching the same dot as from the right approaching 2.5.
So it's clearly approaching the same y-value, regardless of what that y-value
is. And since the question was asking does the limit exist and not asking us
what is limit give me the value of the limit, we are able to answer this
question correctly by saying that yes in fact it does exist. If they asked us what
is the limit and it happened to actually label that Y value right over here as 'a,'
then you can indeed say that the limit as x approaches -2.5
is equal to 'a.' Now one thing I do want to make clear however is this: if
somebody asks you what is f(2.5) basically when x is 2.5
and this is kind of going back to our previous lesson's material just to
refresh your mind ... if somebody asks you what is f(2.5) what should
your answer be? Well you might be tempted to just say 'a' because the left-side
limit and right side limit seems to be the same number.
But remember limits have nothing to do with what the number is when X is 2.5
because in this situation with that empty dot you must say that F(2.5)
is undefined and once again we have highlighted that asking what the
Y value is when X is 2.5 is not the same as asking what Y value
seems to be approached when X is approaching 2.5.