Video Transcript
Discuss the existence of the limit
as 𝑥 approaches four of 𝑓 of 𝑥 given 𝑓 of 𝑥 is equal to six if 𝑥 is less than
four and 𝑓 of 𝑥 is equal to two if 𝑥 is greater than four.
In this question, we’re given a
piecewise-defined function 𝑓 of 𝑥 and we’re asked to discuss the existence of the
limit as 𝑥 approaches four of 𝑓 of 𝑥. And before we recall what we mean
by the existence of a limit, we should start by taking a look at our function around
the value of 𝑥 is equal to four. And there’s a few different reasons
for this. First, there’s lots of different
ways a limit cannot exist. So, instead of recalling all the
different ways a limit might not exist, we should start by taking a look at the
outputs of our function near 𝑥 is equal to four. We can then see what problems arise
when we try and find a value for this limit.
Another reason is our function is a
piecewise-defined function. This means it’s several different
subfunctions defined over several different subdomains. In particular, one problem that can
occur is when our limit point is at the endpoints of the subdomains of our
subfunction. This means that our function is
defined differently to the left and to the right of our limit point. We can see this is what’s happening
in this case. Four is the endpoint of the
subdomains of 𝑓 of 𝑥.
And at this point, we can start to
notice something interesting. When our input values of 𝑥 are
less than four, our function 𝑓 of 𝑥 is outputting a constant value of six. However, when our input values of
𝑥 are greater than four, our function is outputting a constant value of two. So it appears our function has
different left and right limits as 𝑥 approaches four. We can use this to discuss the
existence of this limit.
We recall that we say the limit as
𝑥 approaches 𝑎 of a function 𝑓 of 𝑥 exists and is equal to a finite value of 𝐿
if the limit as 𝑥 approaches 𝑎 from the right of 𝑓 of 𝑥 and the limit as 𝑥
approaches 𝑎 from the left of 𝑓 of 𝑥 both exist and are both equal to 𝐿. In other words, for the limit as 𝑥
approaches 𝑎 of 𝑓 of 𝑥 to exist, we need to check three things. First, we need to check that the
limit as 𝑥 approaches 𝑎 from the right of 𝑓 of 𝑥 exists. Second, we need to check that the
limit as 𝑥 approaches 𝑎 from the left of 𝑓 of 𝑥 exists. Third, we need to check that both
of these are equal to the same value of 𝐿.
But we can see this is not what’s
happening in this case. For example, we could do this using
a diagram. However, we can actually do this
directly by using limits. Let’s start by finding the limit as
𝑥 approaches four from the left of 𝑓 of 𝑥. Since we’re taking the limit as 𝑥
approaches four from the left, our values of 𝑥 are always less than four. And in this case, our function 𝑓
of 𝑥 is equal to a constant value of six. Therefore, this limit is just equal
to the limit as 𝑥 approaches four from the left of six. But six is a constant, so its value
doesn’t change as our value of 𝑥 changes. So this limit evaluates to give us
six.
We can do exactly the same to
determine the limit as 𝑥 approaches four from the right of 𝑓 of 𝑥. This time, our input values of 𝑥
are greater than four. So our function 𝑓 of 𝑥 is equal
to the constant value of two. Therefore, this limit is just equal
to the limit as 𝑥 approaches four from the right of two. And two is a constant, so this
limit just evaluates to give us two. So the left limit was equal to six
and the right limit was equal to two. These are not equal. So we can say that the limit as 𝑥
approaches four of 𝑓 of 𝑥 does not exist.
And in particular we can use the
notation 𝑓 evaluated at four from the left to represent the limit as 𝑥 approaches
four from the left of 𝑓 of 𝑥 and 𝑓 evaluated at four from the right to represent
the limit as 𝑥 approaches four from the right of 𝑓 of 𝑥. We can then conclude the limit does
not exist as 𝑓 evaluated at four from the left is not equal to 𝑓 evaluated at four
from the right.
It’s worth noting we can also check
this answer by considering a graph of 𝑓 of 𝑥. 𝑓 of 𝑥 is a constant value of six
when 𝑥 is less than four and a constant value of two when 𝑥 is greater than
four. So it’s the horizontal line 𝑦 is
equal to six when 𝑥 is less than four. And it’s the horizontal line 𝑦 is
equal to two when 𝑥 is greater than four. This gives us the following. We can then see the left and right
limit from the diagram. As 𝑥 approaches four from the
left, our outputs are a constant value of six. So the left limit is six. However, as our values of 𝑥
approach four from the right, our outputs are a constant value of two. So our limit as 𝑥 approaches four
from the right is two. These are not equal. So we can also conclude that the
limit as 𝑥 approaches four of 𝑓 of 𝑥 does not exist because 𝑓 evaluated at four
from the left is not equal to 𝑓 evaluated at four from the right.
