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## Calculus, all content (2017 edition)

### Unit 1: Lesson 16

Limits of piecewise functions

# Worked example: point where a function is continuous

AP.CALC:
LIM‑2 (EU)
,
LIM‑2.A (LO)
,
LIM‑2.A.2 (EK)
Sal finds the limit of a piecewise function at the point between two different cases of the function. In this case, the two one-sided limits are equal, so the limit exists.

## Want to join the conversation?

• at he substitutes x with 3 but the function is defined between 0 < x < 3. In other words it's defined up to three but not three. So isn't that technically wrong?
• in that particular function you mentioned,, it is continuos until x approaches 3.
We needed to find the limit of f(x) of that function as x approached 3. That is only the idea of limit.. It is not defined at x =3, but we can find a value of f(x) so that it would have formed a continuos graph.. So we find the value of f(x) that would have been for x=3.

Hope that was not complicated
• At how we can understand that function is continuous?
• It is not undefined for any positive argument. This means that there are no asymptotes or removable discontinuities, but proving continuity can be done in a variety of ways (for instance, noting that it is differentiable or noting that its inverse is differentiable etc.). Since differentiability is a stronger condition than continuity, all differentiable functions are also continuous over the differentiable interval.
• Can someone please suggest me a video in which all log functions are explained? Because I am only aware of the basic stuff but I guess as we proceed, we need to know the complicated functions of log too!
• What is a piecewise function?
• A piecewise function has different rules in different intervals. For example, look up aat this function:

f(x) = x^2 if x if x<4
= 4 if x<4 or x=4

Between the interval wich goes from negative infinity, it is x^2; and between the interval wich goes from 4 to positive infinity it is always four.

To give a counterexample, g(x)=x^2+1 is not a piecewise function, because it is always equal to x^2+1; without mattering the value of x
• At and , what if the function is non continuous? What numbers do we plug in for x?
• If there is a jump discontinuity, then the limit from the left side and the limit from the right side will not be equal so the overall limit does not exist. You still have to plug in the same x value in both equations and you will get different values so the overall limit does not exist. But if there is a removable discontinuity, both the limit from the left side and the limit from the right side will be equal so the overall limit exists. In any case, you have to plug in the same x value.
• I forgot what a log is :/
• A logarithm is essentially the opposite of the exponential function. What this means is that if a^x = b, the log(base a) b = x.
• Why didn't we find the function value when x =3 to check if that is equal to the limit to satisfy the condition of continuity
• for a limit to be continuous, lim(x tends to c) f(x) = f(c).

IN this case we know that lim(x tends to 3) g(x) = log(9) but we don't know if
g(3)=log(9).

So how can we say that the limit is continuous
• We do know that g(3)=log(9), because the function g is defined at x=3 and we can plug 3 into the function.

g(3) = (4-3)*log(9) = 1*log(9) = log(9)
• I'm a bit confused by the title; this only proves that the limit of g(x) as x -> 3 exists, not that g(x) is necessarily continuous at that point right?

EDIT: nvm guys we used direct sub, so it is indeed continuous
• for a limit to be continuous, lim(x tends to c) f(x) = f(c). so we first see if the limit exists and then we see if it's equal to the function of that point, at this example we are not examining if the function is continuous it is continuous, he is showing us the case.
• How would I explain why g(x) is continuous?

## 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.