hi everyone today we're going to talk about how to prove that a limit does not exist to
complete this problem we'll exam each part of the definition of the limit and then look
for the part of definition that is not true for our function let's take a look in this
particular problem we've been asked to prove that the limit does not exist and the limit
that we're trying to prove does not exist is the limit as x approaches zero of the absolute
value of x divided by x now if you remember the definition of a limit says that the limit
exists if the left hand limit exists the right hand limit exists and the two the left and
right hand limits are equal to one another so if we're going to prove that the limit
does not exist we either have to prove that the left hand limit does not exist and or
that the right hand limit does not exist and or the two are not equal to each other so
what we'll do is start with the left hand limit make sure we've got one then make sure
that we've got a right hand limit and then make sure they are equal to one another and
if we can't pass any of those steps then we'll have proven the limit does not exist so let's
start with the left hand limit of this function we'll take the limit as x approaches zero
from the negative or left hand side of the absolute value of x divided by x we'll also
be we'll just go ahead and write it now we'll also be trying to find the limit as x approaches
the zero from the positive or right hand side of the absolute value of x divided by x so
if you can imagine here with the left hand limit call this left hand limit and this the
right hand limit with the left limit we're approaching zero from the left hand side so
if we have our number line here and we're approaching zero we have these other points
negative three negative two negative one and we're coming in from this side approaching
zero so imagine if we plug in negative three here for x for the left hand limit we plugged
in negative three we would get and let's just say this we plugged in negative three we'd
get the absolute value negative three divided negative three which would give us on top
here the absolute value of negative three is a positive three divided by a negative
three and we would get negative one you can see if we plugged in negative two right we'd
get the absolute value of negative two divided by negative two it's going to give us two
over negative two and again we get negative one so what we start to realize is that no
matter what value we plug in because we're always plugging in a negative value and these
absolute value bars will always turn that negative value into a positive value we'll
have a positive over a negative the same value and therefore we'll always end up with negative
one so what we realize here we can really simplify this into x divided negative x is
always what we're going to end up with and of course that simplifies to negative one
so we can really call this here the limit as x approaches zero from the negative or
left hand side of negative one if we plug in zero right if we plug in the value we're
approaching zero we're still just going to get negative one because we can't even plug
in our value here we just always have negative one so that limit is going to be negative
one and we can see that we do in fact have a left hand limit that it does exist so now
we have to go and look at the right hand limit well we're going to end up with a similar
thing over here if we imagine the values we have over here the values one two and three
and we're coming in from this side approaching zero if we plug in three or two or one we're
always going to have down here you know three over three and even if we put these absolute
value bars on it we're still just going to get positive three over three which is going
to give us one if we plug in two we'll get the same thing we'll always have a positive
value over a positive value and they're going the numerator and denominator will be equal
to one another so we're always going to get one and we can write that in the same way
we would say that we'll always end up with x over x which is always going to give us
one and we can call that the limit as x approaches zero from the positive or right hand side
of one will always give us one because again plugging in zero to this function here that
we've simplified it to one we can't even plug in zero it's always just going to be one but
this shows us that we do in fact have a right hand limit and that it does exist so remember
that the definition of the limit is the left hand limit exists the right hand limit exists
and the two are equal to one another we'll we've shown that we have a left hand limit
right our left hand limit is negative one we've also shown that we have a right hand
limit our right hand limit is equal to positive one so we've gotten two thirds of the way
through the definition of the limit but of course the third piece of our limit definition
is that the two are equal to one another and obviously our right hand limit is not equal
to our left hand limit and so we say that the limit does not exist because the left
hand limit does not equal the right hand limit because negative one does not equal positive
one and so we've proven that the general limit the limit as x approaches zero of the absolute
value of x divided by x that that limit does not exist because of this fact here that the
left and right hand limits are not equal to one another so i hope you found that video
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