Start  The topic for today is limits and this is a critical piece for understanding
calculus calculus is built on the idea of limits well actually they develop
calculus before they know how to do limits but that they were a lot of
mistakes so in order to make everything work right
the idea of limits was introduced to make sure that there was everything was
on a solid foundation now one of our motivating examples for calculus is the

0:30  idea of position or distance traveled vs velocity and so if you know for example
that you go on a car trip and you travel 100 miles in two hours then we get an
average of 50 miles per hour over that will period but perhaps in the first part you

1:02  want a little slow and you went 20 miles in the first hour and the second hour you
manage to get on the expressway and you went 80 miles in the second actually maybe we
did it the other way around you want eight miles in the first hour and then
you got pulled over and you got a ticket you were stopped trouble there and then the second hour

1:34  you only went 20 miles because you spent a lot of time sitting by the side of the
road while the guy was writing you a ticket so here we have a very difference
in velocity and we averaged over smaller times we get different rates say if we
look more closely at what you were doing during this middle . you were in fact
going 0 miles for half an hour and so your average speed is zero and we look on

2:08  smaller and smaller and smaller range as we get some time
is very different very different speeds
if we look at this in terms of a graph in the first hour you didn't go anywhere
at all but you went there quite then you started going quite quickly then you got
pulled over and you didn't go anywhere for a while and then you sped up and you

2:33  went somewhere now a question we might ask is how fast were you going right at
the moment when you got pulled over and the way that we would have to figure
that out or is by averaging the distance over smaller and smaller intervals or
zooming in on this graph and looking very closely and what's going on and
then we compute the slope of the tangent line but in order to do that computation

3:00  we may have to divide by a very small time interval and we want to know what
happens as we let that time interval go smaller and smaller and smaller so
instead of looking at this distance versus velocity problem what we can do
is look at just the general question of suppose that I have some function that I

3:32  can't compute at some specific moment but i can compute values nearby in this
case this would be I can look at how fast you go how far you travel in any
finite time interval whether that's one hour 1 minute 1 second one tenth of a
second but i want to know what happens as we get very close so instead of
taking the situation of speed vs distance i'm going to just take a general

4:04  function so for example suppose we look at some function like f of X is sine 3x
divided by X this is a perfectly good function except if X is
not00 this problem this function has a problem at zero because certainly
absolute 0 is 0 / 0 which is undefined

4:40  we don't know what number we can assign to that but if we look at this function
for values very close to zero so for example if we look at f of x computed
some if we look at f of 0.2 we get a perfectly good number of 2.8 23

5:03  look at f of 0.1 we get another nice number of f of 0 (can't read my notes)
f of 2.9552 f of 0.01 is 2.99 2.9887 and so and if we look at smaller and

5:39  smaller numbers we get lots of 9's here it looks like the answer is getting very
close to three
similarly if we look at negative numbers to look negative numbers and s0 0.2 its
the same as that because the sine of negative 0.6 is whatever it is we're

6:05  dividing a negative x negative we get the same answer so this is exactly the
same if we look on either side of zero
and if we draw the graph this function will get something that looks roughly

6:32  like this this is PI negative PI and so on so we'll get a nice function and it
sure looks like we want this value to be three that doesn't mean that f of 0 is 3
what it means is as we look closer and closer and closer but not exactly at 0
the value gets closer and closer and closer to three which is what we see

7:04  here by plugging in so we would say in this case  this is not a proof, this is
just a justification. To actually prove that sin(3 x)/x is actually 3 is a little challenging,
we will do it but not yet  but we would say in this case if we could justify
this completely; I can we do it later  is that the limit as X goes to 0 of

7:33  sin(3x) divided x is in fact three. In this notation each of
these little symbols mean something here we write limit to mean that we're taking the limit
and this X arrow 0 means that X is
closer to closer and closer to 0 but it never is 0 so we have this in general if

8:06  we write something like the limit as whatever variable we choose let's call
it Y now and it goes to some number like 5 of some function let's call an H and
we say it's some number like 10 so suppose we have this statement limit y
goes to

8:30  five h y is 10 this means the words as y get's closer and closer to
five hy gets closer and closer to10
it doesn't mean that h of x or h of y is

9:11  10 we haven't defined h5 we don't care what the value is but it doesn't mean
that it gets really close we can say that in a very precise way but let me
hold off on that a little bit okay let me do one other example where we can
actually convince ourselves other than just by looking at the graph we can do

9:31  some algebra
let's do another example where we convince ourselves that this statement
is truly true (truly true?) also true
suppose we have some function let's call it G of X which looks like 1 minus x
divided by 1 minus the square root of x and i want to know what does G look like

10:02  near x=1 it's not defined at x equals 1 g 1 11 / 11 which is 0 / 0 and this is
a problem g equals want one is not in the domain of this function this
function is defined for all positive numbers except well actually all

10:32  nonnegative numbers except for x equals 1
so how can we figure this out again we can draw a graph that's one thing we can do
and so if i asked my computer to draw the graph it draws me a nice graph of
this function when x is 0 we get 1 over 1 which is one and then as x increases
the function increases here at x equals 1 we don't have a value and then it

11:03  carries on nicely after that and it looks kind of like a square root
function and this value here if we plug in a bunch of numbers let me not you can
we plug in a bunch of numbers will see that this gets very close to two it's
not too because it's not defined when x is one but it's close to two now I want
to claim here that in fact the limit as X goes to one of G of X is in fact too

11:38  and I want to do more than just say it's true saying is true is fine you know
what this means now I hold means we know that it gets really close here but let's
confirm that it's true
so here this to confirm that this is true i'm going to do some algebra with
this and check if I can do some algebraic simplification to make this

12:04  look a lot like something that I can do so if i take 1 minus x + 1 minus square
root of x so this is G of X it's this and here i'm going to assume X is not
one ever forbid so i'm going to assume this X is never one this is my
hypothesis here because I want to calculate the limit as X is near one but

12:35  X is never allowed to be one so i can assume X is not one I take this and I'm
going to do some algebra and I don't like this square root in the bottom so
what I'm going to do to this is I'm going to do a magic trick
x 1 minus 1 plus the square root of that over 1 plus the square root of x that's
just one back to make it clear it's just I'm going to x 15 x 1 nothing has

13:03  changed but i'm going to write one in a weird way i'm going to write one as 1
plus the square root of x divided by the square of X X is not one
well it doesn't matter here and now when i do the algebra out on the top i get 1
plus x times 1 is 1 sorry 1 minus X plus square root of x minus x squared x

13:40  divided by 1 and the middle term drops out because I get a minus square root X
+ + square root X and they cancel and then square root of x times square root
X is X so now i have to do a little more notice that this is 1 minus X and this

14:05  is the square root of x times 1x plus the square root of x times 1x
and on the bottom I still have 1x
oops I'm walking off the screen. so i have 1 minus X plus square root of

14:32  x times 1 minus x divided by 1 minus X and i can simplify that by factoring 1x
on each of these terms and on the bottom i still have one X so this is
that but this is not one so if X is not 1 1 minus x over 1 minus X make sense

15:08  and it is 1 so that means that this is just 1 plus the square root of x so
after all of this algebra blah blah
what have we done shown that the following: (this is g(x)) if x is not one
then g of x is exactly the same as 1 plus the square root of x so important

15:41  to remember as long as X is not one these are the same function as long as X
is not one but this one's easy
now that means that if we take the limit
as X goes to one of G of X that will be
exactly the same thing as the limit as X goes to 1 of 1 plus the square root of x

16:04  because they are exactly the same as long as X is not one and when we're
taking women we don't care what happens at the value only here this is just too
so that means that we've actually proven we've actually made sure that the limit
as X goes to one of G of X 2 ok they're not always this hard. I picked a

16:44  hard one on purpose to show you that we can do hard ones.
