today I'm going to help you to understand how to prove the limit as X approaching Infinity of
this thing is equal to 2/3. yes we are talking about proving a limit rigorously (must know for college calculus and real analysis) but this time
we'll actually be using the epsilon-N definition because we have X is approaching Infinity so let
me just give you guys a quick review whenever we're proving limits in fact pretty much these
four versions if x is approaching some number a and we have some function f(x) suppose this
is equal to some finite number L well in this case we'll be using the classic Epsilon Delta
definition and we have done a lot of examples on this already so you guys can go check them out
here if x is the approaching a finite number a we'll be using Delta if the
limits finite we'll be using Epsilon you can just import them as notations for now
for our case here we have the limit as X approaching Infinity of sum function f of x
and this is you know 2/3 is finite so let's write it down as L so if we have a finite limit we
will be using Epsilon but if x is approaching Infinity we'll be using capital N so again just
think about this as notation I'm going to explain these in detail in a minute and then the other two
version is that if we have limit as X approaching some number a of sound function well if the limit
is equal to Infinity here we'll be using capital M and here we have X is approaching a final number
so we'll be using Delta so this is the n Delta definition lastly the limit as X approaching
Infinity of some function f of x and let's say this is equal to Infinity for this one you see
the limit is infinity so we'll use capital M and X is approaching Infinity so use capital N yes
all right and of course we can also have negative Infinity here here and here but you know that will
be for later on so these are the four main versions that you should know today we'll
talk about this one so let me give you guys the definition and then we will write the proof and
then at the end I'm going to use the specific Epsilon and you'll see how everything works
so here we go today we have the limit as X approaching Infinity of some function
and let's say this is equal to a finite limit L this right here means here we go let's review this
again L is finite if the limit is finite then we use Epsilon so for all Epsilon greater than
zero there is sixth this time is X approaching Infinity so use capital n there is some number
capital N that's greater than zero classically you know such that such that what this time we
just care about X going you know all the way to the right If X is bigger than this capital n
so this means if we are going far far away you know toward the right if x is big enough then
the distance between f of x and the limit is going to be close enough so this test again
that's the absolute value and now we just do this minus that so we have f of x minus L and then
they should be close enough that means it's less than Epsilon in fact you can be as close as you
want for this and that so notice this part is pretty much the same thing as what we did
before with the upstand Delta definition but it's just that when we have X is approaching Infinity
we want X to be greater than some capital N all right and one more thing before to the
proof this is very common when we talk about sequence because when we have sequence it only
makes sense for us to take the limit as n goes to infinity and it's like the small n here all
right and then that's the sequence a n okay here we go if you want to write a proof easily these
are the four words to remember all right but before anything of course as always put down
Pierre first and the four words are given and then choose and then suppose and then check
so whenever we have for all apps on greater than zero so we just go ahead and write given
Epsilon greater than zero and you can also say be arbitrary but that's optional in my
opinion this is good enough then whenever it says there exists that means we are going to choose
and and points and we don't know and don't worry about it just write it down right here and this
is perhaps the hardest part all right but for now just write it down and just calm down and
then we proceed now we want this to happen right so suppose we have X is greater than n so that's
the third part suppose X is big enough right bigger than this capital N then we want to have
this right so this is implication if then assuming we have this then we want to have this to happen
so let's check if that part is indeed going to happen and that's
absolute value this minus that the function minus the limit so 2x plus 1 over 3x plus 4 minus two
star and hopefully we can work this out and show that this is less than Epsilon and it will be done
so this is usually the blueprint for the definitions like this for the first few things
that we do is just algebra so that's the fun part just go ahead and combine the fractions and I'm
just going to multiply this and that so we have 3 times 3x plus 4 and now we'll multiply this and
that which is 6X plus 3 and then multiply this and that so 3x times negative 2 so negative 6X
and then four times that which is negative eight and I will still keep the absolute value notice
6X minus 6X cancel and then we have 3 minus eight which is negative five by itself the absolute
value you can just write it as positive 5 right and then just take out the absolute value and then
over the street doesn't matter right with positive so just drag it without the absolute value
and now be careful with this we have 3x Plus 4. technically we should put absolute value right but
here's the thing if x is greater than n and here's the thing if x is approaching Infinity that means
X is positive right right here we can really say greater than zero both of them are positive right
so in fact this is pretty good because 3 times x this is positive and then we add another positive
number everything's positive absolute value doesn't matter so I'll just write it as 3x Plus 4.
sometimes we may end up with a minus here then in that case you may still have to keep the absolute
value for a little bit or you will just have to put down some more condition right here
okay now what do we do after all the algebra well now it's a good idea to use the condition that we
have and that is X is greater than n here's the thing let's go ahead and replace x with n
so let me just write it down here 5 over 3 times 3 capital n and then plus four and let's think about
it what's the connection between this and that X is bigger than n that means this denominator
is bigger than this denominator meaning this whole fraction will be smaller than this whole fraction
and this is the inequality that we like because we want to show that this is less than Epsilon
right so we are on the you know right track now here are two ways that you can proceed
the first way is the more beautiful way in my opinion is that you can go ahead and set this
5 over 3 times 3 M plus four because we really want to make this equal to Epsilon right and
that was Epsilon so let's just go ahead and set it to be Epsilon and then solve for n
that's okay because remember n is going to be expression in terms of Epsilon and later on
you can also see that and it's going to be positive because Epsilon is positive
this is good but let me tell you sometimes when we are doing Epson Delta have more Epsilon and
definition as long as you know how to work with this inequalities then it's actually pretty easy
you don't have to go through this way here's the easiest way for this check this out so
imagine that if we don't have the plus four
what does that mean that means this thing so check this out I'm just going to write down 5
over 3 times 3M which is 9N all right and I'm supposed to have three times four um which is
plus 12 but I'm aiming knowing that you see this denominator is bigger than that that means what
that means the whole fraction is going to be again smaller than this of course the 5 stays
so we can just kind of take away the four and then just say this thing is smaller than that perfect
and now what do we do well you can again just go ahead and put five over nine and set it to
be Epsilon and then figure out what n is but again we can play around with this inequality
check this out 5 over 9 is less than one so I can just write this as less than 1 over capital n
yes and then finally I'm just going to write this down right here one over and
put it to be Epsilon that means n is going to be one over Epsilon
so right here I'm just going to say this is equal to 1 over n is 1 over Epsilon and then
right now we can actually go back here I'm just going to choose n to be 1 over Epsilon
have a look because Epsilon is greater than zero so that means one over Epsilon has to be
greater than zero so this has to be true right I'm going back and you know just write down everything
and what's one over one over Epsilon of course this is equal to Epsilon
and water [Music] and that's that's how I do it that's it so if you just want to see how to draw a
nice proof like this then you are done you're done seriously now let me take some time to explain
this in detail as I promised I will give you a specific Epsilon and then we'll see how 6 works so
suppose I'm going to pick Epsilon to be 0.02
so what does that mean though here let me give you guys a picture I'm going to grab this function for
you guys so it will look like this right here and I'm just going to care about the first quadrant
all right and when X is equal to zero we have one over four so I take this and then we have this
increasing but it's going to have a horizontal asymptote at two thirds so this right here
is 2 3. Epsilon is 0.02 and right here absolute value of the function minus the limit again we
want the function to be really really close to the Limit and this time the error was the distance
from the limit is 0.02 and it's just going down because the function is only down below right so
I just have to worry about coming down here and of course this is natural to scale 0.02 is really
small but you know the idea if I have two over three minus 0.02 like this right here all right
this is going to be right here again the blue dash line is L and then the red dashed line is L minus
Epsilon as you can see the intercept right here and this is going to be the capital N that we have
to find all right so how cool we make it happen well when we have actual numbers we can actually
just do some algebra and this is not so bad this is the curve that we have over there so all I have
to do is solve the equation 2x plus 1 over 3x plus 4 by the way one two three four like this number
you put the function equal to this number um just say 2 over 3 minus 0.02 right and then just go
ahead and solve for x and then you are going to get the answer for this right here X is equal
to I'll just show you guys the answer for this this right here is 97 over 150 right so you can
do it whichever way that you want but X is going to be 238 over 9.
and this right here is approximately so I'll just give you guys approximation which is about
26.4 all right and that's public the entropy so this is 26. 4 now if you're working with little n
right here if you're talking about sequences then you should pick capital n as the whole number but
right here we're talking about X and the capital N is just any number so you can say 26.4 each
and you are pretty much done again this end will work this is the smallest possible end to make
this Epsilon um work well if this statement work now earlier we found out that we have a formula
to compute the end and that is n equals 1 over Epsilon so what does that mean though if Epsilon
is equal to 0.02 that means our capital N based on our computation earlier is just 1 over 0.02 and it
just work that out we know that's equal to 15. and you see this demo right here says if x is greater
than n then this should happen if x is greater than 50 that's somewhere right here if x is
greater than 50 of course it's greater than 26.4 the curve is going to be in between of this that
and you might be wondering is this 3D legit yes it is whenever you are doing the rigorous
proof for limits as known as the inequality work cell then nobody can get mad at you all right so
that's it and I also want to talk about earlier I told you guys that um we could have looked at
this right here and then set out to be Capital the center to be Epsilon and figure for capital
N is right so let's go ahead and do this in blue real quick so if we have five over I mean purple
five over three times three n plus four and then if we put this to be Epsilon let's just solve
this so I'm going to multiply the reciprocal of this so we get 1 over 3 n plus four equals
three Epsilon over five and then I'm going to take the reciprocal on both sides so 3 capital N plus
4 equals five over three Epsilon and I'm going to bring the four to the other side and then divide
the three on both sides so capital N will be 5 over divided three again nine and then UPS down
and then four it will be a minus four here over this three so this n you could have put this right
here earlier and now work perfectly as well except for one little thing notice we have a minus 4 over
street so that means this end might be negative sometimes so in this case you could also use right
you could also use and because I want to avoid this being negative so what we can do is just
go ahead and pick the max amount of zero or this expression 5 over 9 Epsilon minus four over three
that means if this happens to be negative just choose zero and that works perfectly
and if you use this formula right here with this Epsilon guess what yes you are going to end up
with this for your answer check this out I'm just going to plug in 0.02 so n will be equal to
5 over 9 times 0.02 and then minus 4 over 3 work that out all right you will really get 238 over
9. multiply this by three and three and into workable fractions so that's it seriously if
you just kind of calm down and just try to understand all the um for all their exists
and other stuff all the rigorous definition of Demands this can prove are really really fun