hi everybody you know for this kind of question all we have to do is to put a
2 into the x and then we are done so the answer is 1 over 2. yay yay
ok we are aldualts now so for this kind of questions please don't say put a 2 for x and
say the answer is equal to 1 over 2 and then box it and then go home no don't do that because how
do we know that this is indeed correct yes we will have to do the proof for it yes we'll have
to use the epsilon delta definition for limits and yes it's on my shirt already also on the back
let me remind you guys what that is okay so the idea is that if we have the limit as
x approaching some number a of a function f of x if we do have the limit to be l this right here
means the following first we say for all epsilon greater than zero okay so let me just put that
down right here then there must exist a delta greater than zero with the following property
so i just put down such that here's the deal if the distance between x and a is small enough
well distance means the absolute value and we just go ahead and subtract x and a like so
how small is model smaller than delta and right here we don't want the x to be equal to a so we
look at this backwards and say this is greater than zero if this happens then we must have
the distance between f of x and the limit l it's also small enough we write this down
absolute value of the function f of x minus the limit l we want this to be less than epsilon
this is crazy there's nearly no numbers except for zero and the rest are just
english alphabets and also greek letters crazy huh but don't worry i'm going to make this easier for
you guys with four key words to do this kind of questions but first off i'll tell you whenever
we have to write proof just go ahead and put down pf your professor your teachers will be
really happy when they see pa for proof trust me all right anyway the definition star wars
for all epsilon greater than zero so the first word that you are using is
given all right given and then just write this down given epsilon greater than zero
you can put it down i guess and some people will say you should write it like this let epsilon
be greater than zero b r b tree well this is also okay and in fact i would say that this right here
will make you sound more like a mathematician but this right here is just as fine shorter of course
but anyway either way it's okay first word given next it says there exists a delta greater
than zero when there's exists when there is there exists we are going to find it so we'll say choose
the outer to be something we don't know yet just go ahead and leave it for now and then we'll come
back to it a lot of students they get like stuck right here because we don't know how to put down
right here so we just stuck but no don't worry let's just go ahead and continue we'll come back
to this later and everything will be okay so continue we look at this condition we want to
have this condition so we are going to say suppose so this is the third word that you should remember
suppose this right here so let's just go ahead and write that down absolute value x and then the end
here is two so we will just have x minus two like so suppose this right here is less than the ota
and then again we don't want x to be equal to 2 so we say this right here is greater than 0 like this
lastly we want this to happen that we don't know yet so let's just go ahead and say we
are going to check if that will indeed happen or not you can also say we see so that it's like okay
we are going to see that it will happen or not but check is also okay so forwards
given choose suppose check and let's just go and write this down we have the absolute value
the function is x over one one over x what x one over x and then minus one over two
all right so this is the part that we have to do the most amount of work and the first thing
that we'll do is just some algebra so we can get a common denominator right here so multiply 2 and 2
x and x so all together we will get absolute value 2 minus x over 2x and then right here we can just
switch the order of subtraction and the reason is because we have this the absolute value of x minus
2. so we will have the absolute value and then to switch the order of subtraction we'll just have
to factor out negative and then we will have the x minus 2 inside like this and then over 2x now
the absolute value of a quotient is a quotient of the absolute values so we can just go ahead
and say absolute value of negative x minus 2 over absolute value of 2x and guess what the
absolute value it's a negative here the negative doesn't matter so we can just say bye bye to it
and let's go ahead and write that down first so we have the absolute value
of x minus two and i'm going to write this down next like this times one over
the absolute value of 2x is the same as the absolute value of 2 times the absolute value of
x and the absolute value of 2 is just 2 and then we have the absolute value of x that's so cool
now this is really good because we have the assumption that the absolute value of x minus
2 is less than delta so we can go ahead and say this is less than delta now the hard part is
what do we do with this our goal is to find a bound for this expression and hopefully we can
say that this right here is less than something if we can do that then everything will go smoothly
so now this is how we are going to do it come back to here we have to have some more things to work
with let's just go ahead and write down we want the minimal and then put on like a brace like so
and right here you can pick any number that you want as long as it's greater than zero
don't be too crazy just use one you can put on one half you can put on 17 up to you all right
one is okay let's just use one the moment you do that we will be able to come here and just look
at this part of the inequality and say that the absolute value of x minus 2 is less than
delta which we have the one right here we can just say 1 like so and then based on this we will
somehow get an expression of the 1 over absolute value of x from here all right so let's just
go ahead and open the absolute value we'll get the absolute value of x minus 2 must be between
negative one and one and then you see that the itself the absolute value is just x
so right here let's just go ahead and add two add two and add two
so we will get one and then we have the x in the middle and then one plus two which is
the essay at one i meant to say add two add to add two but anyway one plus two is three
now this is a pretty common mistake i will say some people may say we need the absolute value
of x so they look at this and then they just take the absolute value of x right away and then they
see that one and three three is bigger so they just put absolute five x is less than three
okay from here to here it's okay nothing wrong with that but the thing is that we
want to have 1 over absolute 5 of x so we just do the reciprocal on both sides
since they are both positive when we do the reciprocal we will have
to switch the inequality and unfortunately we get greater than and then one over three
if we come here we don't want this to be greater than one over three that's not what we want
like what i said earlier we want to have an upper bound for this right we want to say this
is less than something so unfortunately this right here is not the way to go
you know it happens just go ahead and erase the mistake and then just remember right this
is exactly what we need so it was not like an innocent mistake that was a horrible mistake we
will actually have to fix that but it's not a big problem because all we have to do is look at this
and do the reciprocals first so 1 over 1 is 1. reciprocal of that is 1 over x and the
reciprocal of that is 1 over 3 and they are all positive so we'll just flip the inequality like so
now we do the absolute value the absolute value of 1 over x is the
same as absolute value of 1 which is just 1 over the absolute value of x like this
which one's bigger one is bigger so we can say this right here is less than one yeah
less than that's exactly what we want and also we see that we have to have a two on the bottom
so let's just go ahead and multiply 1 over 2 on both sides ladies and gentlemen from here
we know 1 over 2 times the absolute value of x is less than 1 over 2.
therefore we can come back here and say that this is less than which we have that already
one over two very good yeah okay we're not done yet but almost
the idea is that you see we have this expression it's less than yeah it's older
stuff and it's less than and we want this to be less than epsilon so that's our final goal
okay here's the question if we want to end up with epsilon what's your delta b
so the question is what times 1 over 2 will give us epsilon well the answer is what just
two epsilon isn't it because we see if we have two epsilon times one over two this is perfect because
the one half and the two will cancel yeah so we will just go ahead and come back here and say
delta should have that two epsilon let's just go ahead and put that in the minimum here
oops now three this is the backwards absolutely no no no just two epsilon like this yeah and
then close that and then because right here we're seeing the minimal you see right here
we're missing a symbol we will use less than or equal to because we're the minimum so as
we can see we have this expression is less than epsilon so thanks to the epsilon theta definition
in fact we are done and finally we can just draw a box and shade this e
adult calculus what do you guys think if you want to learn more calculus then one good place for
you to start is our sponsor today brilliant it is an excellent online learning platform with a
big focus on interactive learning as we can see this is from their calculus in a national course
this course uses visual and physical intuition to present the major topics of calculus namely
limits derivatives integrals and infinite series you will walk away with a clear sense of what
calculus is and what you can do i can tell you that taking a course from brain is really really
fun because they will always keep you engaged by storytelling and beautiful animations besides
calculus they also have other courses in math science and computer science so you have a lot
of choices to choose from and i have a special link for you guys use the link in the description
brilliant.org slash blackpenredpen because that way you can get 20 off discount i want to thank
brilliant for sponsoring this video i also want to thank you guys for checking them out