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  • Let's review our intuition of what a limit even is.

  • So let me draw some axes here.

  • So let's say this is my y-axis, so try to draw a vertical line.

  • So that right over there is my y-axis.

  • And then let's say this is my x-axis.

  • I'll focus on the first quadrant,

  • although I don't have to.

  • So let's say this right over here is my x-axis.

  • And then let me draw a function.

  • So let's say my function looks something like that,

  • could look like anything, but that seems suitable.

  • So this is the function y is equal to f of x.

  • And just for the sake of conceptual understanding,

  • I'm going to say it's not defined at a point.

  • I didn't have to do this.

  • You can find the limit as x approaches a point where

  • the function actually is defined,

  • but it becomes that much more interesting, at least for me,

  • or you start to understand why a limit might be relevant

  • where a function is not defined at some point.

  • So the way I've drawn it, this function

  • is not defined when x is equal to c.

  • Now, the way that we've thought about a limit

  • is what does f of x approach as x approaches c?

  • So let's think about that a little bit.

  • When x is a reasonable bit lower than c,

  • f of x, for our function that we just drew, is right over here.

  • That's what f of x is going to be equal. y is equal to f of x.

  • When x gets a little bit closer, then our f of x

  • is right over there.

  • When x gets even closer, maybe really almost

  • at c, but not quite at c, then our f of x is right over here.

  • And the way we see it, we see that our f of x

  • seems to be-- as x gets closer and closer to c

  • it looks like our f of x is getting closer

  • and closer to some value right over there.

  • Maybe I'll draw it with a more solid line.

  • And that was actually only the case

  • when x was getting closer to c from the left,

  • from values of x less than c.

  • But what happens as we get closer and closer

  • to c from values of x that are larger than c?

  • Well, when x is over here, f of x is right over here.

  • And so that's what f of x is, is right over there.

  • When x gets a little bit closer to c, our f of x

  • is right over there.

  • When x is just very slightly larger than c, then our f of x

  • is right over there.

  • And you see, once again, it seems

  • to be approaching that same value.

  • And we call that value, that value that f of x

  • seems to be approaching as x approaches

  • c, we call that value L, or the limit.

  • And so the way we would denote it

  • is we would call that the limit.

  • We don't have to call it L all the time,

  • but it is referred to as the limit.

  • And the way that we would kind of

  • write that mathematically is we would say the limit of f

  • of x as x approaches c is equal to L.

  • And this is a fine conceptual understanding of limits,

  • and it really will take you pretty far,

  • and you're ready to progress and start thinking

  • about taking a lot of limits.

  • But this isn't a very mathematically-rigorous

  • definition of limits.

  • And so this sets us up for the intuition.

  • In the next few videos, we will introduce

  • a mathematically-rigorous definition

  • of limits that will allow us to do things

  • like prove that the limit as x approaches c

  • truly is equal to L.

Let's review our intuition of what a limit even is.

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A2 初級 美國腔

16-1 限制的正式定義第 1 部分:直覺審查(Formal definition of limits Part 1: intuition review | AP Calculus AB | Khan Academy)

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    yukang920108 發佈於 2022 年 07 月 05 日
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