Video Transcript
Find the limit as 𝑥 approaches
four of 𝑓 of 𝑥 if 𝑓 of 𝑥 is equal to five plus 𝑥 squared plus three 𝑥 over the
absolute value of 𝑥 plus three if 𝑥 is greater than negative four and less than
zero and 𝑓 of 𝑥 is equal to five 𝑥 plus four if 𝑥 is greater than zero and less
than four.
In this question, we’re given a
piecewise-defined function 𝑓 of 𝑥 and asked to determine the limit as 𝑥
approaches four of 𝑓 of 𝑥. And to answer this question, we
need to start by noticing one thing. We’re asked to find the limit as 𝑥
approaches four of 𝑓 of 𝑥, and four is one of the endpoints of the subdomains of
our piecewise function 𝑓 of 𝑥. And since this is the endpoint of
one of the subdomains, this means our function 𝑓 of 𝑥 will change definition to
the left and to the right of four. So we need to consider the left and
right limits of 𝑓 of 𝑥 as 𝑥 approaches four.
We can do this by recalling the
following property. We say the limit as 𝑥 approaches
𝑎 of 𝑓 of 𝑥 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. And at this point, we can notice
something interesting. We can find the limit as 𝑥
approaches four from the left of 𝑓 of 𝑥, since our function 𝑓 of 𝑥 is equal to
the linear function five 𝑥 plus four when 𝑥 is between zero and four. So when we take the limit as 𝑥
approaches four from the left of 𝑓 of 𝑥, we can choose our values of 𝑥 to be
closer and closer to four. So we can choose them between zero
and four. This means its limit is the same as
the limit as 𝑥 approaches four from the left of five 𝑥 plus four.
And we can then evaluate this limit
by direct substitution. We substitute 𝑥 is equal to four
into the near function. We get five times four plus four,
which is equal to 24. So the left limit exists and is
equal to 24. However, none of this was
necessary. We can notice directly from the
definition of 𝑓 of 𝑥 that the limit as 𝑥 approaches four from the right of 𝑓 of
𝑥 does not exist. And this is a direct consequence
from the fact no input value of 𝑥 greater than four is in the domain of 𝑓 of
𝑥.
And it might be useful to just find
the domain of 𝑓 to be safe. We can do this by finding the union
of its subdomains. This gives us the open interval
from negative four to four, where we remove the point zero. And we can see that no value
greater than four is in this set. So we can’t take the limit as 𝑥
approaches four from the right of 𝑓 of 𝑥.
Therefore, we can say that the
limit does not exist because the limit as 𝑥 approaches four from the right of 𝑓 of
𝑥 also doesn’t exist.
