Ok, so day two. Finishing up the notes. Now that we know how important limits are to Calculus,
let's get the notation out of the way and do some examples of finding limits using a
graphical approach and numerical approach where we make some t-tables. So limit notation,
let me get out of the way here. f is a function defined on some open interval containing the
number a. Now, this value of a. We are going to be allowing our x values to approach a
particular value "a". The limit is going to be the y value that the graph approaches.
The function f may or may not be defined at a. You are going to see this through some
examples. At the value of a we may have a hole where the graph approaches and the graph
is undefined at that hole, but yet it is approaching the same point from both sides. Thus we will
still be able to find the limit. Sometimes we won't be able to find the limit. We will
explain why as we get to the end of this board. A function's graph could be continuous or
have a hole like I just said at a or be broken at a. Now the notation is the limit as x approaches
a, we are going to let our x values again approach some particular value of a. The limit
of a function is equal to L. It is read again, the limit of f(x) as x approaches a is L.
Now this value of L is a y value. A real limit, when you look at these graphs as examples,
the limit of a function is the y value that graph approaches. We will be approaching particular
a from the left and we will have a one sided limit on the left side. We will be approaching
the value of a from the right and have a one sided limit coming from the right. Then we
will decide if they are equal. Let me get out of the way and do that. We can approach
from the left. That is denoted by an exponent of negative or just put in a minus sign in
the upper right hand corner. We will have the limit as x approaches a from the left
of f(x) equals some real y value. Or we could approach a from the right and that is denoted
by positive or a plus sign in the upper right hand corner of a where the exponent would
normally be. So we will be able to look at some graphs in a second and look for limits
from the left and the right. If they are equal, then we are going to be able to state that
there is a two sided limit. If they are not equal, well then there is no two sided limit.
Basically in that is what you are most concerned with. Those two sided and so they are kind
of a big deal knowing are the left and the right limit the same. If they are, then the
two sided limit exists. If they are not, then the two sided limit does not exist. Let's
look at some examples. (one sided limits do have applications) Yes, it is another day
and I am still making this video. So let's do five examples and get this done. Here I
have got f(x) is equal to negative x squared plus 5x plus 6. This is a parabola opening
down with the negative leading coefficient. It is a polynomial. There is one variable
with positive whole number exponents. So, it is a smooth polynomial function. Here is
the parabola opening down. I want to do these five examples in, like I stated in the beginning
of the video, a graphical and numerical approach. Then the next video is going to be doing limits
using the properties and algebra. I already have the parabola drawn for us. I want to
look the value of the actual function and values of the limits. So f(1) is equal to
ten. Now whether I take one and plug it into the equation, or I just look at my graph at
the x value of 1 which is this value. One is my value of 'a' that I was talking about
in my previous page of notes. If I go to my x value of one and I go up to the graph, it
is equal to ten. Now limits though are the y value of the function or the y value the
graph approaches as you approach 'a'. So we are going to take this value of one and approach
it again from the left and the right. We are going to look at the graph and we are going
to approach that x value of one from the left hand side or from the x values that are smaller
than one. So we are going to come up and you can see that it is approaching a y value of
ten. Yes, this graph is defined everywhere so it actually...the y value will actually
be ten at one. But the limit as you approach one from the left and you are like at .9999999999
it is going to be like 9.9999 something. As we approach it from the right, it kind of
actually goes above ten but then you come back down and approach ten...the y value of
ten...from both the left side of one and the right side of one. So the left limit is going
to be equal to ten. The right side limit with the positive little exponent there is going
to be ten. Our left and our right hand side limits, left and right, are the same values.
When your left and right side limits are the same, or your one sided limits are equal the
two sided limit exists and thus the limit of this function is equal to ten at x equal
1. Over here, we have a little bit of a different situation. I have already worked out the algebra.
I didn't want to just completely give the picture without the equation because we are
going to in the next section doing these algebraically and I would like to remind you where holes
come from in a graph. When your denominator of a rational function is equal to zero, or
can be equal to zero, you get things like vertical asymptotes often. Sometimes you will
have... If your numerator's degree is larger than the denominator's degree by one you are
going to have a slant asymptote. When the zero is able to be cancelled out like we have
here, we of f(x) is equal to x squared plus x minus 6 over x minus 2. But if I take the
numerator and I factor it, the x-2 cancels out. So, this reduced version of the equation
f(x) is equal to x+3. That is like saying y equal mx plus b. That is an equation of
a line. So, this graph or this function crosses the y axis at three, has a slope of one so
from the y value of three up one and over one, up one and over one, and so on. But,
since this was my original function and it was undefined at the value of two we get this
hole. Now my textbook, most of these questions or half of the questions just give the picture
and asks you to analyze the picture to find the limit. That is the graphical approach
of finding limits. So we have a little bit of algebra and graphing review in the picture.
So looking at this as my original function because it is, what is f(2)? F(2), if I take
2 and I plug it into my equation, as I can see from my graph we have a hole, the function
there is undefined. That is what I meant in my notes when I said the function could be
broken or have a hole. The function may not be defined at this value of a that you are
trying to find the limit at. Indeed we are going to find the limit at 2. Now the graph
is undefined at two, but as you approach 2 from the left the graph is again coming up.
You are using values...you are looking at x values that are less than 2 to the left.
So coming in from the left we are approaching a y value of one, two, three, four, five.
As you come in from the left we are getting ever increasingly close to five. So, you know
this function is not undefined at 1.999999...., it is only undefined at that exact value of
two. So as you approach 2 from the left your y value becomes or approaches 5. As you approach
from the right, as you come in from the right, it is a hole so it is going to be...the function
or graph is going to come to that hole approaching that same y value both sides. Actually from
this way. So it is going to approach 5 as you come in from the right. Then since your
left and right limits are the same, your two sided limit exist which is 5. Again, the limit
is not 5 every where, the limit as x approaches 2 from the left and the right...and thus since
they are equal on both sides...the left and the right limits are the same on both sides,
the limit at 2...the limit as x approaches 2 not at 2 but as x approaches 2 from the
left and the right, you limit is equal to 5 even though the actual function at 2 is
undefined. My next two examples are going to be pure graphical. I am not going to include
equations in the last example, the fifth example is going to be using a table. Ok, I said I
was not going to include the equations. But as I am erasing the board I remember one of
my best students just told me they were still struggling a little bit with piecewise functions.
So, we have f(x) is equal to x-3 and f(x) is equal to 4. x-3 is going to be used for
x values that are less than negative one. At y equals 4 or f(x) equals 4 is for when
the x values are greater than or equal to negative one. Now what I have done here is
dotted in the two equations. This has a slope of 1 and a y intercept of negative three.
So a y intercept of -3 and slope of 1. This is of course, if you look at f(x) as just
y, y is equal to 4 is a horizontal line. I have made a horizontal line through y equals
4. But, in a piecewise function you don't just graph a bunch of equations or functions
on one x y axis. You have to pay attention to when do you want to use those. I only want
to look at the line f(x) is equal to x minus 3, or y=x-3, for x values that are less than
negative one. I am going to come through here at negative one, I am going to find that line
and I am going to put an open dot there and highlight the part of the graph where my x
values are less than negative one. I am getting rid of the rest of this dotted line. The second
equation or the second part of my piecewise function is f(x) is equal to 4, or y=4, a
horizontal line. I want to use that function for x values that are greater than or equal
to negative one. I am going to come down here to negative one again, go to the graph and
this time make a solid dot indicated by or needed because of that equal sign in my inequality.
I am highlighting the part that is to the right. This is my piecewise function. We are
going to answer some questions graphically. What is f(-1)? At negative one my actual function
value is equal to 4. The limit as I approach negative one from the left, the little negative
exponent, as I come in from the left side of negative one my graph is down here. I am
approaching, my y values are approaching or my limit is approaching...as x approaches
negative one getting closer and closer to a y value of negative one, negative two, negative
three, and negative four. As I approach negative one from the right, so here I am to the right
of negative one...my x values value greater than negative one, I am up here and all of
my y values are equal to 4. So as I approach negative one from the right, all my values
are actually at already and staying at a y value of four. My right hand limit is positive
four. Now I have been able to find out with this piecewise function a left limit and a
right limit as my x values approached negative one from the left and negative one from the
right. But, my left and right hand limits are not equal. If your left and right limits
are not equal, then the two sided limit does not exist. There is none. Ok. Now over here
I have got another piecewise function. It is a little bit different. I have got f(x)
is equal to 2/3 x minus 1. I have taken the privilege, whatever, I have already dotted
in that line. It has a y intercept of negative one and slope of 2/3. Negative one and then
up two and over three. The dotted line I will fill in in a second. Then I have a got f(x)
is equal to one. Well the first part of this piecewise function is defined everywhere except
for where x is equal to 3. So, as my previous board full of examples had, I have a hole
at an x value of three. But unlike the previous example I had up here a minute ago, I have
another part of this piecewise function that says that when x does equal 3.... I don't
want it to be the same number. Let's make that 4. When f(x) is equal to 4 when x is
equal to three. When x does equal 3, my actual function value is equal to 4. I have a function
with a hole in it. It has just been pulled out and plugged in somewhere else in this
piecewise function. So f of 3. Well f(3), if I am using an x value of three, then I
am not working with that one. I'm using the second equation. My y value is four. Or when
I go to my graph and I see an x value of three, my dot that is open, it is at again 4. The
limit as we approach 3 from the left is equal to 1. Excuse me. As I approach 3 from the
right, ok I am still approaching a value of one. Again my left and right hand limits are
equal so the two sided limit exists. What if you have a function that maybe you don't
necessarily know what it is going to look like graphically and you don't have a graphing
calculator handy? Last example. Alright, last example. Limit as x approaches zero of sine
of x over x. Now if this were the beginning of Precalculus when we were studying rational
functions all of our problems had polynomials in the numerator and the denominator. If that
were the case and you have a value in the denominator that will allow the fraction to
be undefined, then we often had vertical asymptotes. If the top the degree was larger than the
bottom degree, or the numerator's degree was larger than the denominator's degree by a
value of one then you had a slant asymptote. But this is certainly just a polynomial. We
have a trig function in the numerator and a denominator equal to x. If we just try to
find the limit as x approaches zero, we cannot find this functions value at zero. If you
plug in zero, you are dividing by zero and that is undefined. I don't really know what
this graph looks like. So your other option is to, and it is a bit of a monotonous one,
but to make a t-table effectively, and don't make a t-table like you are trying to graph
it. You are trying to find the limit as x approaches zero. So you need to get very very
close to zero on both the left and the right hand side. Don't just plug in whole number
values like negative two, negative one, not that would even work because you cannot take
the sine of negative two. (yes you could, I am thinking of arcsine) But, when you make
this table of values to find the limit make sure that you say very very close to that
'a' value where are allowing x to approach both from the left and from the right. So
as we can see, I can't do this in my head so I have already found these answers, as
we approach a value of zero from the left hand side my y values are getting and they
are already very close to 1. That is just what this function looks like and these values
are close to zero. They are all less than zero, so they are approaching zero from the
left. Make sure that you get closer in value...to that 'a' as you approach from the left. As
we approach from the right .03, a little close at .02, and getting closer with .01 you can
see that my y values are getting very very close to 1 coming up from the same direction
from both sides...getting close to 1. So the limit as I approach zero from the left of
f(x)....or I should just write the expression sin(x) over x is equal to getting closer to
1. The limit from the right is also equal to one. Thus the two sided limit,
I will write it down here, the limit as x approaches zero for the sine of x over x,
the left and the right hand limits are the same so my two sided limit is equal to one.
If you to actually sketch this out, it would look something like this, maybe not sketch
it but with your graphing calculator you would get something that looks like this. But, it
is undefined at zero so there is a hole there as we discussed. We could not plug in zero.
It made the function undefined. That is your graphical and numerical introduction to finding
limits in Calculus. Next video will be finding these algebraically. BAM!!! Go do your homework:)