hi everyone today we're going to talk about how to use the graph of a function and the
precise definition of the limit to find delta to complete this problem we'll identify the
side of the limit that constrains the definition and then use that value of epsilon to find
delta let's take a look in this particular problem we've been asked to use the graph
of the f of x equals the square root of x to find delta such that if the absolute value
of x minus four is less than delta then the absolute value of the square root of x minus
two is less than zero point four and you may notice that this problem is strongly related
to the precise definition of the limit or the epsilon definition of the limit which
I've written here for reference which says that if zero is less than the absolute value
of x minus a which is also less than delta then the absolute of f of x minus l is less
than epsilon with this problem we've also been given the graph of our function y equals
the square root of x which we've drawn here in orange and this graph is going to be really
helpful to us because it's going to show us or give us a visual representation of what
we're trying to find remember with problems like this we're usually asked to find one
of two things we're either asked to find delta or epsilon if you have to find delta you're
usually given epsilon and you have to find epsilon you are usually given delta in this
case we've been given epsilon right we've been told that epsilon is equal zero point
four and we can see that here epsilon in the precise the definition of the limit in our
problem is zero point four and what that means is that we've been allowed to move a distance
away from our y coordinate here of two a distance of zero point four so we have our y coordinate
here of two and we've been allowed to move point four in this direction and point four
in this direction to get to one point six or two point four and each of these distances
here we can call epsilon this distance is equal to epsilon and this distance is equal
to epsilon they're both equal to zero point four so what this problem is asking us to
do is given that constraint given that epsilon has to be less than or equal to point four
and we have to stay within a distance of point four in either direction away from two what
it's asking us to do is say how far away can we get away from four and still stay within
that range so how far can we get away from four either this distance here or this distance
here how far can we get away from four and still stay within that range well the interesting
thing about this problem is that we can only define one value for delta right we can see
that the distance from this line to four and the distance from this line to four are very
different and we can only define one value for delta so the problem is if we were to
define this second distance here as delta if we said that this was equal to delta then
that means the we would be allowing ourselves to move an equivalent distance to the left
away from four right let's maybe say that would be about like this the same distance
and that would throw us way outside our epsilon right if we could move all the way to this
point here then that would mean we could move all the way down to here with epsilon and
we can't we can't move lower than one point six so this line here really represents the
farthest that we can move away from four so this distance right here has to define delta
and then the equivalent distance we'll say we can only move about this far to the right
of four and we'll have to stay within this range here that distance that delta will insure
that we'll stay within our epsilon range over here on the y axis so given that that means
that we are much more interested in this point here and how far that is away from x equals
four than we are in this point so we're going to try to figure out how far this point is
away from the point x equals four notice that that point corresponds to the y value of one
point six so thats going to be the value the y value that we're interested in given that
that's the y value that we're interested in we can just go ahead and plug that straight
into our function we have the function f of x equals the square root of x and we know
that f of x or y is equal to one point six so we can plug one point six in for f of x
and set that equal to the square root of x now we just want to solve for x so we'll go
ahead and square both sides when we take one point six by one six or one point six squared
we get two point five six equal to x what this tells us is that the corresponding x
value for y value of one point six is x equal to two point five six so this point right
here is at two point five six so now what we really need to know the distance between
two point five six and four because that will give us our delta here so we want to say four
minus two point five six to find the difference between the two and when we do that arithmetic
we get one point four four so what that tells us is that delta must be less than or equal
to one point four four if delta is less than or equal to one point four four then what
we're saying is that we'll never move greater than one point four four away from x equals
four either to the left or to the right and if we never move farther away from x equals
four then one point four four we will stay within this range here we'll stay within this
boundary and this boundary and that guarantees that we'll stay within this boundary one point
six and up here within two point four if we had gone out to this line here if we had gone
out to this line then we would have blown out of our one point six six epsilon boundary
on the y axis so I hope you found that video helpful if you did like this video down below
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