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  • - [Voiceover] We have the graph of y is equal to g of x

  • right over here, and what I wanna do is I wanna check

  • which of these statements are actually true

  • and then check them off.

  • And like always, I encourage you to pause the video

  • and see if you can work through this on your own.

  • So let's look at this first statement.

  • So this first statement says both the limit of g of x

  • as x approaches six from the right-hand side

  • and the limit as x approaches six from the left-hand side

  • of g of x exist.

  • All right, so let's first think about the limit

  • of g of x as x approaches six from the right-hand side,

  • as we approach six from values greater than six.

  • So if we look over here, we could say okay,

  • when x is equal to nine, and g of nine is right over there,

  • g of eight is right over here,

  • g of seven is right over here,

  • looks like it's between negative three and negative four,

  • g of 6.5 looks like it's a little bit,

  • it's still between negative three and negative four

  • but it's closer to negative three.

  • G of 6.1 is even closer to negative three.

  • G of 6.01 is even closer to negative three,

  • so it looks like the limit

  • from the right-hand side does exist.

  • So it looks like this one exists.

  • Now let's see, and I'm just looking at it graphically.

  • That's all they can expect you to do

  • in an exercise like this.

  • Now let's think about the limit as x approaches six

  • from the left-hand side.

  • So I could start anywhere, but let's say when x is equal

  • to three, g of three is a little more than one.

  • G of four looks like there's a little bit less than two.

  • G of five looks like it's close to three.

  • G of 5.5 looks like it's between five and six.

  • G of 5.75 looks like it's approaching nine.

  • And as we get closer and closer,

  • as x gets closer and closer to six from below,

  • from values to the left of six,

  • it looks like we're unbounded, we're approaching infinity.

  • And so technically, we would say this limit does not exist.

  • So this one does not exist.

  • So I won't check this one off.

  • Some people say the limit is approaching infinity,

  • but that technically is,

  • infinity is not a value that you can say

  • it is approaching in the classical formal definition

  • of a limit.

  • So for these purposes, we would just say

  • this does not exist.

  • Now let's see, they say the limit as x approaches six

  • of g of x exists.

  • Well, the only way that the limit exists

  • is if both the left, if both the left, the left,

  • and the right limits exist and they approach the same thing.

  • We'll, our limit as x approaches six from the negative side

  • or from the left-hand side, I guess I could say,

  • does not even exist.

  • So this cannot be true.

  • So that's not gonna be true.

  • The first one's not gonna be true.

  • G is defined as x equals six.

  • So at x equals six, it doesn't look like g is defined.

  • Looking at this graph, I can't tell you

  • what g of six should be.

  • We have an open circle over here, so g of six

  • does not equal to negative three,

  • and this goes up to infinity,

  • and we have a vertical asymptote actually drawn

  • right over here at x equals six.

  • So g is not defined at x equals six.

  • So I'll rule that one out.

  • G is continuous at x equals six.

  • Well, you can see that it goes up to infinity

  • then it jumps down, back down here, then it continues.

  • So when you just think about it in commonsense language,

  • it looks very discontinuous.

  • And if you wanna think about it more formally,

  • in order for something to be continuous,

  • the limit needs to exist at that value.

  • The function needs to be defined at that value,

  • and the value of the function needs to be equal

  • to the value of the limit

  • and neither of these, the first two conditions are true

  • and so these can't even equal each other because neither

  • of these exist.

  • So this is not continuous at x equals six

  • and so the only think I can check here is none of the above.

  • Let's do another one of these.

  • So the first statement, both the right-hand

  • and the left-hand limit exists as x approaches three.

  • So let's think about it.

  • So x equals three is where we have this little

  • discontinuity here, this jump discontinuity.

  • So let's approach, let's go from the positive,

  • from values larger than three.

  • So when x is equal to five, g of five is a little bit

  • more negative than negative three.

  • G of four is between negative two and negative three.

  • G of 3.5 is getting a bit closer to negative two.

  • G of 3.1, it's getting even closer,

  • closer to negative two.

  • G of 3.01 is even closer to negative two.

  • So it looks like this limit right over here,

  • well, I'm circling the wrong one,

  • it looks like this limit exists.

  • In fact, it looks like it is approaching negative two.

  • So this right over here is equal to negative two.

  • The limit of g of x as x approaches three

  • from the right-hand side,

  • and I'll just think about it from the left-hand side.

  • So I can start here.

  • G of one, looks like it's a little bit

  • greater than negative one.

  • G of two, it's less than one.

  • G of 2.5 is between one and two.

  • G of 2.9 looks like it's a little bit less than two.

  • G of 2.99 is getting even closer to two.

  • G of 2.99999 would be even closer to two

  • so it looks like this thing right over here

  • is approaching two.

  • So both of these limits, the limit from the right

  • and the limit from the left exist.

  • The limit of g of x as x approaches three exists.

  • So these are the one-sided limit.

  • This is the actual limit.

  • Now in order for this to exist, both the right

  • and left-handed limits need to exist

  • and they need to approach the same value.

  • Well, this first statement,

  • we saw that both of these exist but they aren't approaching

  • the same value.

  • From the left, we are, or sorry,

  • from the right, we are approaching,

  • we are approaching negative two.

  • And from the left, we are approaching two.

  • So this limit does not exist.

  • So I will not check that out or I will not check that box.

  • G is defined at x equals three.

  • Well, when x equals three, we see a solid dot

  • right over there.

  • And so it is indeed defined.

  • It is indeed defined there.

  • G is continuous at x equals three.

  • Well, in order for g to be continuous at x equals three,

  • the limit must exist there.

  • It must be defined there,

  • and the value of the function there

  • needs to be equal to the value of the limit.

  • Well, the function is defined there,

  • but the limit doesn't exist there

  • so it cannot be continuous.

  • It cannot be continuous there.

  • So I would cross that out.

  • And I can't click, I wouldn't click none of the above

  • because I've already checked something,

  • or I've actually checked two things already.

- [Voiceover] We have the graph of y is equal to g of x

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A2 初級

Worked example: Continuity at a point | Limits and continuity | AP Calculus AB | Khan Academy

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    yukang920108 に公開 2022 年 07 月 05 日
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