Name: 1.55 極限在不連續的點（Limit at a point of discontinuity）
Uploaded: 2022-07-02T03:01:00.000Z
Duration: 6 min 20 s
Description: 在不連續點處限制

祈使句型

Let's say that f(x) is equal to the absolute value of x minus three over x minus three

and what I'm curious about is the limit of f(x) as x approaches three

情態助動詞 A2

and just from an inspection you can see that the function is not defined when x is equal to three - you get zero over zero: it's not defined

So to answer this question let's try to re-write the same exact function definition slightly differently

So let's say f(x) is going to be equal to - and I'm going to think of two cases:

I'm going to think of the case when x is greater than three

So when x is - I'll do this in two different colors actually

When x - I'll do it in green - that's not green

When x is greater than three, what does this function simplify to?

Well, whatever I get up here, I'm just taking the..

I'm going to get a positive value up here and then I'm...

Well, if I take the absolute value it's going to be the exact same thing, so let me...

For x is greater than three, this is going to be the exact same thing as x minus three over x minus three

because if x is greater than three, the numerator's going to be positive, you take the absolute value of that, you're not going to change its value

so you get this right over here or, if we were to re-write it...

...if we were to re-write it, this is equal to, for x is greater than three, you're going to have f(x) is equal to one

Similarly, let's think about what happens when x is less than three

When x is less than three, well, x minus three is going to be a negative number

When you take the absolute value of that, you're essentially negating it

so it's going to be the negative of x minus three over x minus three

or if you were to simplify these two things, for any value as long as x doesn't equal three

this part right over her simplifies to one, so you are left with a negative one

I encourage you, if you don't believe what I just said, try it out with some numbers

Any number greater than three, you're going to get one

You're going to get the same thing divided by the same thing

you're going to get negative one no matter what you try

This is my f(x) axis - y is equal to f(x)

and what we care about is x is equal to three

so x is equal to one, two, three, four, five

and let's say this is positive one, two, so that's y is equal to one

this is y is equal to negative one and negative two

So this way that we have re-written the function

we've just written [it] in a different way

but if our x is greater than three, our function is equal to one

so if our x is greater than three, our function is equal to one

It looks like that, and it's undefined at three

and if x is less than three our function is equal to negative one

so it looks like - I'll be doing that same color

Well, let's think about the limit as x approaches three

from the negative direction, from values less than three

...as x approaches three, the limit of f(x)...

...as x approaches three from the negative direction

and all this notation here - I wrote this negative as a superscript right after the three - says

Let's think about the limit as we're approaching...

Let's think about the limit as we're approaching from the left

as we get closer and closer and closer...

So, say we start at zero, f(x) is equal to negative one

We go to one, f(x) is equal to negative one

We go to two, f(x) is equal to negative one

If you go to 2.999999, f(x) is equal to negative one

So it looks like it is approaching negative one if you approach..

...if you approach from the left-hand side

...the limit of f(x) as x approaches three from the positive direction, from values greater than three

So here we see, when x is equal to five, f(x) is equal to one

When x is equal to four, f(x) is equal to one

When x is equal to 3.0000001, f(x) is equal to one

We seem to be approaching a different value when we approach from the left

and if we are approaching two different values then the limit does not exist

So this limit right over here does not exist

(Let me write this in a new color - I have a little idea here)