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Course: AP®︎/College Calculus BC > Unit 1
Lesson 11: Defining continuity at a point- Continuity at a point
- Worked example: Continuity at a point (graphical)
- Continuity at a point (graphical)
- Worked example: point where a function is continuous
- Worked example: point where a function isn't continuous
- Continuity at a point (algebraic)
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Worked example: point where a function is continuous
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.
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at2:05he 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?(21 votes)
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(28 votes)
At1:45how we can understand that function is continuous?(11 votes)
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.(5 votes)
- 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!(9 votes)
I forgot what a log is :/(5 votes)
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.(5 votes)
why did he add 10 there suddenly??(5 votes)
In math, "log" is short for "logarithm". It's a way to show how many times we need to multiply a number (the base) to get another number. When we say "log" without specifying a base, it's assumed to be base 10. That's a common math practice. So, he wasn't adding 10. He was explaining that "log" means "log base 10".
I recommend going back to Algebra 2 if you're still confused on logarithms: https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs(4 votes)
- What is a piecewise function?(4 votes)
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(6 votes)
- At1:47and3:16, what if the function is non continuous? What numbers do we plug in for x?(4 votes)
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.(6 votes)
- 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(3 votes)
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)(4 votes)
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(3 votes)
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.(3 votes)
- 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(4 votes)
2:05
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.