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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 of X is equal to the absolute value of X minus 3 over X minus 3 and what I'm curious about is the limit of f of X the limit of f of X as X approaches 3 and just from an inspection you can see that the function is not defined when X is equal to 3 you get 0 over 0 it's not defined so to answer this question let's try to rewrite the same exact function definition slightly differently so let's say f of 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 3 and when X is less than 3 so when X is good just in two different colors actually when x do it in green it's not green when X is greater than 3 when X is greater than 3 what does this function simplify to well whenever I get up here I'm just taking the I'm going to get a positive value up here and then I'm what if I take the absolute value it's going to be the exact same thing so let me 4x is greater than 3 this is going to be the exact same thing as X minus 3 over X minus 3 because if X if X is greater than 3 the numerator is 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 rewrite it if we rewrite it this is equal to 4 X is greater than 3 you're going to have f of X is equal to 1/4 X is greater than 3 similarly let's think about what happens when X is less than 3 when X is less than 3 well X minus 3 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 3 over X minus 3 or if you were to simplify these two things for any value as long as X does not equal 3 this part right over here simplifies to 1 so you're left with a negative 1 negative 1 for X is less than 3 I encourage you if you don't believe what I just said try it out with some numbers try out some numbers 3 point 13.00 1 three point five four seven any number at 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 one you're going to get negative one no matter what you try so let's visualize this function now so now let me draw some axes that's my x-axis and then this is my this is my f of X axis y is equal to f of X and what we care about is X is equal to three so X is equal to one two three four five 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 1 and negative two and we can keep going so this this way that we've rewritten the function is d is the exact same function as this we've just written in a different way and so what we're saying is is where our function is undefined at 3 but if our X's are greater than 3 our function is equal to 1 so if our x is greater than 3 our function is equal to 1 so it looks like it looks like that and it's undefined at 3 and if X is less than 3 our function is equal to negative 1 so it looks like let me do that same color it looks like this it looks like looks like this once again it's undefined at 3 so it looks like that so now let's try to answer our question what is the limit as X approaches 3 well let's think about the limit as X approaches 3 from the negative direction from values less than 3 so let's think about first the limit the limit as X approaches 3 as X approaches 3 and the limit of f of X as X approaches 3 from the negative direction and all this notation here I wrote this negative is a superscript right after the 3 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 if we get if we start with values lower than 3 as we get closer and closer and closer so say we start at 0 the f of X is equal to negative 1 we go to 1 f of X is equal to negative 1 we go to 2 f of X is equal to negative if you go to two point nine nine nine nine nine nine f of 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 of X the limit of f of 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 of X is equal to one when X is equal to four f of X is equal to one when X is equal to three point zero zero zero zero zero zero one f of 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 then when we approach from the right and if we're approaching two different values then the limit does not exist so this limit right over here does does not does not exist or another way of saying it the limit the limit the limit of let me write this in a new color I have a little idea here the limit of the limit of an of a function f of X as X approaches some value C is equal to L if and only if if and only if the limit of f of X as X approaches C from the negative direction is equal to the limit of f of X as X approaches C from the positive direction which is equal to L this did not happen here the limit when we approach from the left we was negative 1 the limit when we approach from the right was positive 1 so we did not get the same limits when we approach from either side so the limit does not exist in this case
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