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  • - [Voiceover] Let f of x be equal to negative one

  • over x minus one squared.

  • Select the correct description of the one-sided limits

  • of f at x equals one.

  • And so we can see, we have a bunch of choices

  • where we're approaching x from the right-hand side

  • and we're approaching x from the left-hand side.

  • And we're trying to figure out do we get unbounded

  • on either of those, in the positive,

  • towards positive infinity or negative infinity.

  • And there's a couple of ways to tackle it.

  • The most straightforward, well,

  • let's just consider each of these separately.

  • So we can think about the limit of f of x

  • as x approaches one from the positive direction

  • and limit of f of x as x approaches one,

  • as x approaches one from the left-hand side.

  • This is from the right-hand side.

  • This is from the left-hand side.

  • So I'm just gonna make a table and try out some values

  • as we approach, as we approach one from the different sides,

  • x, f of x, and I'll do the same thing over here.

  • So, we are going to have our x and have our f of x

  • and if we approach one from the right-hand side here,

  • that would be approaching one from above,

  • so we could try 1.1, we could try 1.01.

  • Now f of 1.1 is negative one over 1.1 minus one squared.

  • So see this denominator here is going to be .1 squared.

  • So this is going to be, this is going to be 0.01,

  • and so this is going to be negative 100.

  • So let me just write that down.

  • That's going to be negative 100.

  • So if x is 1.01, well, this is going to be

  • negative one over 1.01 minus one squared.

  • Well, then this denominator this is going to be,

  • this is the same thing as 0.01 squared,

  • which is the same thing as 0.0001, 1/10000.

  • And so the negative one 1/10000

  • is going to be negative 10,000.

  • So, let's just write that down, negative 10,000.

  • And so this looks like, as we get closer,

  • 'cause notice, as I'm going here I am approaching one

  • from the positive direction,

  • I'm getting closer and closer to one from above

  • and I'm going unbounded towards negative infinity.

  • So this looks like it is negative infinity.

  • Now we can do the same thing from the left-hand side.

  • I could do 0.9, I could do 0.99.

  • Now 0.9 is actually also going to get me negative 100

  • 'cause 0.9 minus one is going to be negative .1

  • but then when you square it the negative goes away

  • so you get a .01 and then one divided by that is 100

  • but you have the negative, so this is also negative 100.

  • And if you don't follow those calculations, I'll do it,

  • let me do it one more time just so you see it clearly.

  • This is going to be negative one over,

  • so now I'm doing x is equal to 0.99,

  • so I'm getting even closer to one,

  • but I'm approaching from below from the left-hand side.

  • So this is going to be 0.99 minus one squared.

  • Well, 0.99 minus one is, is going to be negative 1/100,

  • so this is going to be negative 0.01 squared.

  • When you square it the negative goes away

  • and you're left with 1/10000.

  • So this is going to be 0.0001

  • and so when you evaluate this you get 10,000.

  • So that, or sorry, you get negative 10,000.

  • So in either case, regardless of which direction

  • we approach from, we are approaching negative infinity.

  • So that is this choice right over here.

  • Now there's other ways you could have tackled this

  • if you just look at, kind of,

  • the structure of this expression here,

  • the numerator is a constant,

  • so that's clearly always going to be positive.

  • Let's ignore this negative for the time being.

  • That negative's out front.

  • This numerator, this one is always going to be positive.

  • Down here, we're taking at x equals one,

  • while this becomes zero and the whole expression

  • becomes undefined, but as we approach one,

  • x minus one could be positive or negative

  • as we see over here, but then when we square it,

  • this is going to become positive as well.

  • So the denominator is going to be positive

  • for any x other than one.

  • So positive divided by positive is gonna be positive

  • but then we have a negative out front.

  • So this thing is going to be negative

  • for any x other than one,

  • and it's actually not defined at x equals one.

  • And so you could, from that, you could deduce,

  • well, okay then, we can only go to negative infinity

  • there's actually no way to get positive values

  • for this function.

- [Voiceover] Let f of x be equal to negative one

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A2 初級 美國腔

13-3 分析無限極限:有理函數(Analyzing unbounded limits: rational function | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 05 日
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