Alright in this video they ask us to take the limit as x approaches 2 of the absolute
value of x minus 2 divided by x minus 2 now this is a very common limit type of problem
you will see lots of different exercises that have the absolute value some expression divided
by the same expression and I just want to show you what to do here now we can evaluate
this limit 3 different ways we can evaluate it numerically which is what we will do first
and we can evaluate it graphically by putting it on an x y axis and the coordinate system
and graphing it and seeing what the limit is or we can do it analytically and the way
we will do that is write it as a peace wise function and go that route but we are going
to start with looking at it numerically numerically means we are going to put it in a table so
here I�ve pulled up my TI here and I went ahead an plotted it or entered it in the y
equals mini here and we can go to second table set and if you want to automatically generate
the x y values you can just set both of these to auto and then make it start at 2 and then
just set the width of each individual x value to be like .1 or .01 or something very small
but instead I�m going to make it ask me what I want the x to be and let it automatically
generate it with the y as y is your dependent variable because it depends on your choice
of x which is the independent variable so let�s go to the table and we will just pick
some values around 2 we will pick like 1.7, 1.8, 1.95 it doesn�t have to be anything
specific just pick values very close to 2 we will try it at 2 now you see at 2 there�s
and error right we knew that actually before we started the problem cause if you try and
take 2 and plug it in the expression you get zero divided zero that�s an indeterminate
form but after 2 we have like 2.05, 2.1, 2.2 any values very close to 2 but larger than
2 here�s what we get from our table from our x y table cut that out so I can blow it
up for you see here we go alright so here we go so before 2 it looks the values are
a negative 1, negative 1, negative 1 and above 2 it looks there are positive one, positive
one, positive one now why would that be the case well let�s just think about this logically
for a minute if you pick a value larger than 2 it doesn�t matter how large but just any
value larger than 2 then the difference of x minus 2 will be positive right because x
is bigger than 2 right based on your choice but the absolute value of a positive number
stays positive in which case it will be the same the exact same as the denominator so
you might get 5 over 5, 6 over 6, 7 over 7 so anything you plug in larger than 2 will
give you one now anything less than 2 this difference here will be negative which means
the denominator will be negative but the numerator will be the same numerical value except positive
so those would cancel but would leave you with negative one so what does that seem to
indicate about the limit there is not one definitive numerical value that the function
approaches from both sides of 2 so this will be an example where we say the limit does
not exist now we use this expression so often a lot of limits do not exist that we typically
abbreviate this capital DNE which obviously stands for does not exist now just in closing
I will show you a couple of other things as well if we wanted to do this graphically instead
of numerically we could do the same thing we could either graph this on or TI and you
can kind of see the graph done here at the bottom here I will graph it right here from
the left of 2 is negative 1 and from the right of 2 is positive 1 and so we see the y values
do not go to the same place or to say it another way the limit from the left hand side does
not match the limit from the right hand side so the overall limit would not exist so we
have this sort of graph going on here open dot at 1, at 2 of the y value of 1 going to
the right then an open dot at 2 of the y value of 1 going to the left so then we see that
the limit does not exist.