Left and Right Limits
In this notes we’re talking about one sided limits and when we say one sided limits we mean left and right hand limits or the limit of a function as we approach from the left or negative side or the right or positive side.
So let’s distinguish between one sided limits and the general limit at any given point.
Let’s say we want to take the limit of this function as x approaches the number 4 so maybe let’s say
the limit as x goes to four of the function f of x that we’ve been given.
So square root X times quantity 5 minus X..
So we want to take the limit of this function as x goes to 4.
If you see this we’re talking just about the general limit as x approaches 4 but that’s different than
the left hand limit and the right Him limit.
So if we want to say the left hand limit as x goes to for what we do is we say the limit as x approaches
4 from the left hand or negative side and we write this little negative sign here in the place of like
an exponent so we’re saying as x goes to four from the negative side or the left hand side.
So if you think about the x axis over here we have this point X equals four right here.
So this is the point that we’re approaching the value of x that we’re approaching we can approach it
from the negative side of the x axis or the left hand side so if we’re coming in this way we’re moving
left to right we’re coming from the negative side toward X equals 4.
That’s the limit as x approaches 4 from the negative side or the left hand limit.
And we could take that limit of the same function and that would be different than the limit as x approaches
4 from the positive side of the same function.
If we approach from the positive side we’re approaching from the right hand side because this is the
positive end of the x axis.
So we’re approaching from the positive and we’re approaching from the right.
We’re moving right to left.
So that’s the right hand limit.
So we have three different limits.
All as we get close to this value x equals four.
And again this second limit here this is the lefthand limit or the limit as we approach from the left
hand side.
This is the right hand limit or the limit as we approach from the right hand side.
This first limit is the general limit and it’s important to remember that the general limit cannot possibly
exist in less three conditions are met.
The left hand limit exists number one.
Number two the right hand limit exists.
And number three the left and the right hand limit are equal to each other.
Now when we’re talking about general limits like this oftentimes we just take for granted before we
learn about left and right hand limits are one sided limits.
We just take for granted that the general limit exists without going through the definition.
But truly the general limit can’t exist unless the left hand limit and the right hand limit both exists
and the values of these limits are equal to one another.
If those three things are all true then the general limit can exist and we can find the value of the
general limit the limit as x approaches 4.
Now if we want to take a graphical look at what we’re talking about here let’s go ahead and graph this
function f of x we have f of x equals the squared of X times quantity 5 minus X..
We could just plug in a few points to graph this function.
So if x is equal to zero inside the square root here we have 5 minus zero which is 5 5 times 0 we get
zero so we have the point 0 0 on our function f of x.
If we look at X equals 1 here we have 5 minus 1 which is 4 4 times one is for the square root of four
is to.
So we have the Point 1 2 right here.
If we look at the point x equals four we have 5 minus 4 is 1 4 times 1 is for the square to 4 is 2.
We have the point for two and if we look at the point x equals 5 then we can see that we get 5 minus
5 0 5 times 0 0.
So again we have this point zero.
So we could see is that our graph is going to look something like this.
The curve or the half circle that connects all of these points.
If we plugged in a negative value for x heres what we get.
Lets say we take negative 1 we’d have 5 minus and negative 1 which is 5 plus 1 or 6.
So we have 6 times in negative 1 they’d be negative 6 the squared of negative six is something that
we can’t take without imaginary numbers.
So X equals negative one does not exist in the domain of this function which is why we can see that
the graph stops here at X equals zero.
Similarly if we take a value for x that’s greater than 5 like X equals 6 right here.
And we plug that in.
We get 5 minus 6 which is a negative 1 6 times and negative one gives us a negative six.
And again we’re trying to take the square root of negative six squared of a negative number.
Which is why X equal 6 is also not in the domain of the function.
So this is the graph of our function.
It only exists on the interval X equals zero to X equals 5.
So we have a couple of different interesting points.
If we stick with this example here and we’re interested in the limit as x approaches 4.
First let’s deal with the left hand limit as x approaches 4 or the limit as x goes to 4 from the negative
or left hand side.
Well in that case what we’re doing is we’re tracing the graph from left to right as we get close to
this line here.
X equals 4.
So we’re interested in the value of the function as we trace the graph from the left hand side from
left to right and we get close to this line.
So we trace the graph and we get close to this line where we can see is that the value that we’re approaching
is this point right here which we know was the point four to the value of the function as we get close
to this line is 2.
So we can say that the left hand limit here is equal to 2.
Similarly the right hand limit as X gets close to 4.
So as we approach this line X equals 4 from the positive or right hand side.
What we’re doing is we’re tracing along the graph close to it like this.
Getting close to x equals four.
And as we get closer and closer to this point we get closer and closer to this point here for the value
of the function there is to.
So the right hand limit is also two.
So we can say that the left hand limit exists and is equal to to the right hand limit exists and is
equal to two since both the left and the right hand limit exist and they’re equal to one another.
That means that the general limit also exists and the general limit is going to be equal to the value
we found for the left and the right hand limits of the general limit is also 2.
And of course that makes sense because the easiest way to solve a limit is with substitution.
Assuming that the function is continuous at that point and if we were just plug in the value X equals
four we get 5 minus 4 which is 1 4 times 1 is for the square of 4 is to the value of the function there
is 2 so we can say that the general limit is going to be equal to 2.
But what if we’re interested in the limit of this function as x approaches 5.
So the line X equals 5.
Is this line right here.
And we can see that the domain of the function ends at the line X equals 5.
What can we say about the limit of the function at this point.
More specifically the general limit.
The left hand limit and the right hand limit.
Well remember the general limit is only going to exist if both the left and the right hand limits exist
.
So let’s look at those first.
If we look at the left hand limit as x approaches 5.
So we’re getting closer and closer to the line.
X equals 5.
We traced the graph we’re coming from the left hand side.
So we’re coming from the left hand side or moving left to right.
We’re approaching the line.
X equals 5.
Well as we get closer and closer to the line X equals five we get closer and closer to this point here
the point 5 0.
So what we could say is that the limit as x approaches 5 from the negative side or the left hand limit
and X equals 5.
That that is going to be equal to zero because we’re getting closer and closer to the value y equals
zero.
This point right here.
But what about the right hand limit what if we want to know the limit as x approaches 5 from the positive
side.
Well you can see that everything to the right of equals five is not in the domain of this function.
There is no graph of the function to the right of X equals 5.
Therefore we can’t approach X equals 5 along the graph on the right hand side here.
So there is no right hand limit of the function at that point.
So we can say that this limit does not exist and we abbreviate that with DNA.
So the right hand limit as x goes to 5 does not exist even though the left hand limit as x goes to 5
is equal to zero.
So in this case we can say that the general limit the limit as x approaches 5 of our function f of x
is going to be DNA does not exist because even though the left hand limit exists the right hand limit
does not.
And because the right hand limit doesn’t exist the general limit can’t exist either because remember
the general limit only exists if both the left and the right hand limits exist.
And if they’re equal to one another.
So what we can say is that the limit as x approaches 5 of this function does not exist.
If we wanted to take the limit as x goes to 5 the best we can do is say that the left hand limit exists
at the limit as x goes to 5 from the left hand or negative side is equal to zero but the right hand
limit and the general limit don’t exist at that point.
So that’s the difference between a general limit and one sided limits and how to find the one sided
limits of a function at a particular point