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## One-sided vs. Two-sided limits

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# Limits of piecewise functions: absolute value

AP Calc: LIM‑1 (EU), LIM‑1.D (LO), LIM‑1.D.1 (EK)

## Video transcript

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 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 and when x is less 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... 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 for x is greater than three 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 negative one for x is less than three I encourage you, if you don't believe what I just said, try it out with some numbers Try out some numbers: 3.1, 3.001, 3.5, 4, 7 Any number greater than three, you're going to get one You're going to get the same thing divided by the same thing and try values for x less than three: you're going to get negative one no matter what you try So let's visualise this function now So, now you draw some axes... That's my x- axis and then this is my... 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 we could keep going... 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 and we can keep going... So this way that we have re-written the function is the exact same function as this we've just written [it] in a different way and so what we're saying is... is we're... Our function is undefined at three 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 so it looks like... 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 It looks like this... It looks like... Looks like this... Once again, it's undefined at three So it looks like that So now let's try to answer our question: What is the limit as x approaches three? Well, let's think about the limit as x approaches three from the negative direction, from values less than three So let's think about first the limit... ...the limit, as x approaches 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 me make this clear... Let's think about the limit as we're approaching from the left So in this case, if we get closer... If we get... If we start with values lower than three 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 Now let's think about the limit... ...the limit of f(x)... ...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 So it seems to be approaching... It seems to be approaching positive one So now we have something strange We seem to be approaching a different value when we approach from the left than when we approach from the right and if we are approaching two different values then the limit does not exist So this limit right over here does not exist or another way of saying it: The limit... ...the limit of... (Let me write this in a new color - I have a little idea here) ...the limit of a function f(x) as x approaches some value c is equal to L if and only if... ...if and only if the limit of f(x) as x approaches c from the negative direction is equal to the limit of f(x) as x approaches c from the positive direction which is equal to L This did not happen here - the limit when we approached the left was negative one, the limit when we approached from the right was positive one, So we did not get the same limits when we approached from either side So the limit does not exist in this case