### Video Transcript

What can be said of the limit as π₯
approaches negative six of π of π₯ for the function π of π₯ is equal to negative
nine π₯ minus nine if π₯ is less than negative six and π of π₯ is equal to 45 if π₯
is greater than negative six?

The question gives us a
piecewise-defined function π of π₯. And the question wants us to
discuss the limit of π of π₯ as π₯ approaches negative six. At first, we might be worried about
this. Since our limit is as π₯ is
approaching negative six, we can see that our function π of π₯ is not actually
defined when π₯ is equal to negative six. However, we donβt need to worry
about this. When we say that our limit is as π₯
is approaching negative six, we want to know what happens as π₯ gets closer and
closer to negative six. We donβt actually need to know what
happens when π₯ is equal to negative six.

But now, we can see our second
problem. When π₯ is less than negative six,
we have that our function π of π₯ is negative nine π₯ minus nine. But when π₯ is greater than
negative six, our function π of π₯ is the constant 45. So how would we calculate the limit
as π₯ approaches negative six of this function? And we can get around this by
instead of calculating the limit of π of π₯ as π₯ approaches negative six, we can
look at the left-hand and right-hand limit of π of π₯. We say that the limit as π₯
approaches π of π of π₯ exists if the following three properties hold. First, the limit as π₯ approaches
π from the left of π of π₯ must exist. Second, the limit as π₯ approaches
π from the right of π of π₯ must exist. And third, these two limits must be
equal.

If all of these conditions are
satisfied, then our left-hand and right-hand limits are equal to some finite value
πΏ. And we then say the limit as π₯
approaches π of π of π₯ is equal to πΏ. Since we want the limit as π₯
approaches negative six of π of π₯, weβll set π equal to negative six. So we now need to check if each of
these three properties are true. Letβs start by checking the limit
as π₯ approaches negative six from the left of π of π₯. Since weβre taking the limit as π₯
approaches negative six from the left, all of our values of π₯ are less than
negative six. And we can see from our piecewise
definition of the function π of π₯, if π₯ is less than negative six, then our
function π of π₯ is exactly equal to negative nine π₯ minus nine.

So when weβre taking the limit as
π₯ approaches negative six from the left, our function π of π₯ is exactly equal to
negative nine π₯ minus nine. That means that their limits will
be equal in this case. But now, we can see weβre just
trying to evaluate the limit of a linear function. We can do this by direct
substitution. Substituting π₯ is equal to
negative six gives us negative nine times negative six minus nine, which we can
calculate to give us 45. So weβve shown that the limit as π₯
approaches negative six of π of π₯ exists. Itβs equal to 45. We now need to check the limit as
π₯ approaches six from the right of π of π₯. We can do this in a very similar
way. Since π₯ is approaching negative
six from the right, our values of π₯ will be bigger than negative six.

And from our piecewise definition
of the function π of π₯, when π₯ is greater than negative six, our function π of
π₯ is exactly equal to the constant 45. So when π₯ is greater than negative
six, our function π of π₯ is exactly equal to 45. Therefore, their limits when π₯
approaches negative six from the right will be equal. But now, we see weβre just trying
to evaluate the limit of a constant. And we know a constant does not
change as our value of π₯ changes. So the limit of 45 as π₯ approaches
negative six from the right will just be equal to 45.

So weβve shown the limit as π₯
approaches negative six from the right of π of π₯ also exists. Itβs also equal to 45. Our final condition was that our
left-hand limit and right-hand limit were equal. And we showed that both of these
were equal to 45. So our third condition is also
true. And remember, if all of these three
conditions are true, we say the limit as π₯ approaches negative six of π of π₯ is
equal to the shared value of our left-hand and right-hand limit, which in this case
is 45. So not only have we shown the limit
as π₯ approaches negative six of π of π₯ exists, weβve also shown that itβs equal
to 45.