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

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

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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

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

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

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

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and when x is less than three

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

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

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When x is greater than three...

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

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

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

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

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

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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

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

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...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

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for x is greater than three

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

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

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

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

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

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

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negative one for x is less than three

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

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Try out some numbers:

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3.1, 3.001, 3.5, 4, 7

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

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

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and try values for x less than three:

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

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So let's visualise this function now

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So, now you draw some axes...

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That's my x- axis

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and then this is my...

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

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and what we care about is x is equal to three

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

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and we could keep going...

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

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this is y is equal to negative one and negative two

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and we can keep going...

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So this way that we have re-written the function

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is the exact same function as this

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

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and so what we're saying is...

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is we're...

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Our function is undefined at three

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

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

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so it looks like...

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

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

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

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It looks like this...

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It looks like...

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Looks like this...

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Once again, it's undefined at three

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So it looks like that

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So now let's try to answer our question:

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What is the limit as x approaches three?

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

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from the negative direction, from values less than three

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So let's think about first the limit...

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...the limit, as x approaches three...

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

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...as x approaches three from the negative direction

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

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

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...let me make this clear...

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

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So in this case, if we get closer...

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If we get...

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If we start with values lower than three

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as we get closer and closer and closer...

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

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

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

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

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

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

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Now let's think about the limit...

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...the limit of f(x)...

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

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

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

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

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So it seems to be approaching...

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It seems to be approaching positive one

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So now we have something strange

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

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than when we approach from the right

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

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So this limit right over here does not exist

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or another way of saying it:

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The limit...

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...the limit of...

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

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...the limit of a function f(x) as x approaches some value c is equal to L if and only if...

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...if and only if the limit of f(x) as x approaches c from the negative direction is equal to the limit

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of f(x) as x approaches c from the positive direction which is equal to L

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This did not happen here -

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the limit when we approached the left was negative one,

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the limit when we approached from the right was positive one,

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So we did not get the same limits when we approached from either side

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So the limit does not exist in this case