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## Defining continuity at a point

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# Worked example: point where a function is continuous

AP Calc: LIM‑2 (EU), LIM‑2.A (LO), LIM‑2.A.2 (EK)

## Video transcript

- [Voiceover] So we have
g of x being defined as the log of 3x when zero is
less than x is less than three and four minus x times the log of nine when x is greater than or equal to three. So based on this definition of g of x, we want to find the limit of g of x as x approaches three, and once again, this three is right at
the interface between these two clauses or these two cases. We go to this first case when
x is between zero and three, when it's greater than
zero and less than three, and then at three, we hit this case. So in order to find the
limit, we want to find the limit from the left hand side which will have us dealing
with this situation 'cause if we're less than
three we're in this clause, and we also want to find a
limit from the right hand side which would put us in this
clause right over here, and then if both of those limits exist and if they are the same,
then that is going to be the limit of this, so let's do that. So let me first go from
the left hand side. So the limit as x
approaches three from values less than three, so we're
gonna approach from the left of g of x, well, this
is equivalent to saying this is the limit as x approaches three from the negative side. When x is less than three,
which is what's happening here, we're approaching three from the left, we're in this clause right over here. So we're gonna be
operating right over there. That is what g of x is when
we are less than three. So log of 3x, and since this function
right over here is defined and continuous over the
interval we care about, it's defined continuous for
all x's greater than zero, we can just substitute three in here to see what it would be approaching. So this would be equal to
log of three times three, or logarithm of nine, and once again when people just write log
here within writing the base, it's implied that it
is 10 right over here. So this is log base 10. That's just a good thing to know that sometimes gets missed a little bit. All right, now let's think
about the other case. Let's think about the
situation where we are approaching three from
the right hand side, from values greater than three. Well, we are now going
to be in this scenario right over there, so
this is going to be equal to the limit as x approaches three from the positive direction,
from the right hand side of, well g of x is in this clause when we are greater than three, so four minus x times log of nine, and this looks like some type
of a logarithm expression at first until you
realize that log of nine is just a constant, log base 10 of nine is gonna be some number close to one. This expression would
actually define a line. For x greater than or equal
to three, g of x is just a line even though it looks
a little bit complicated. And so this is actually
defined for all real numbers, and it's also continuous for
any x that you put into it. So to find this limit, to think about what is this expression approaching as we approach three from
the positive direction, well we can just evaluate a three. So it's going to be four minus three times log of nine, well that's just one, so that's equal to log base 10 of nine. So the limit from the left
equals the limit from the right. They're both log nine,
so the answer here is log log of nine, and we are done.

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