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  • - [Voiceover] Which of the following functions

  • are continuous at x equals three?

  • Well, as we said in the previous video,

  • in the previous example,

  • in order to be continuous at a point,

  • you at least have to be defined at that point.

  • We saw our definition of continuity,

  • f is continuous at a, if and only if,

  • the limit of f as x approaches a is equal to f of a.

  • So, over here, in this case,

  • we could say that a function is continuous

  • at x equals three, so f is continuous at x equals three,

  • if and only if the limit as x approaches three of f of x,

  • is equal to f of three.

  • Now let's look at this first function right over here.

  • Natural log of x minus three.

  • Well, try to evaluate it, and it's not an f now,

  • it's g, try to evaluate g of three.

  • G of three, let me write it here,

  • g of three is equal to the natural log of zero.

  • Three minus three.

  • This is not defined.

  • You can't raise e to any power to get to zero.

  • You can try to go to, you could say, negative infinity,

  • but that's not, this is not defined.

  • And so, if this isn't even defined at x equals three,

  • there's no way that it's going to be continuous

  • at x equals three, so we could rule this one out.

  • Now f of x is equal to e to the x minus three.

  • Well this is just a shifted over version of e to the x.

  • This is defined for all real numbers,

  • and as we saw in the previous example,

  • it's reasonable to say it's continuous for all real numbers,

  • and you could even do this little test here.

  • The limit of e to the x minus three as x approaches three,

  • well that is going to be, that is going to be

  • e to the three minus three, or e to the zero, or one.

  • And so f is the only one that is continuous.

  • And once again, it's good to think about

  • what's going on here visually, if you like.

  • Both of these are, you could think of them,

  • this is a shifted over version of ln of x,

  • this is a shifted over version of e to the x,

  • and so if we like, we could draw ourselves some axes,

  • so that's our y-axis, this is our x-axis,

  • and actually, let me draw some points here.

  • So that's one, that is one, that is two, three,

  • two, and three, and let's see,

  • I said these are shifted over versions,

  • so actually, this is maybe not the best way to draw it,

  • so let me draw it, this is one, two, three,

  • four, five, and six.

  • And on this axis, I won't make 'em on the same scale,

  • let's say this is one, two, three.

  • I'm gonna draw one, two, three,

  • I'm gonna draw a dotted line right over here.

  • So g of x, ln of x minus three

  • is gonna look something like this.

  • If you put three in it, it's not defined,

  • if you put four in it, ln of four,

  • well, that's gonna, sorry, ln of four minus one,

  • so that's gonna be ln of four minus three,

  • is actually let me just draw a table here,

  • I know I'm confusing you.

  • So, if I say x and I say g of x,

  • so at three, you're undefined.

  • At four, this is ln of one, ln of one,

  • which is equal to zero, so it's right over there.

  • So g of x is gonna look something like, something like that.

  • And so you can see at three,

  • you have this discontinuity there,

  • it's not even defined to the left of three.

  • Now f of x is a little bit more straightforward.

  • If you have, so e to the three is going to be,

  • sorry, f of three is going to be e to the three minus three,

  • or e to the zero, so it's going to be one,

  • so it's gonna look something like this,

  • it's gonna look something,

  • something like, like that.

  • There's no jumps, there's no gaps,

  • it is going to be continuous, and frankly,

  • all real numbers so for sure it's going

  • to be continuous at three.

- [Voiceover] Which of the following functions

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A2 初級

Functions continuous at specific x-values | 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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