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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

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