episode two limits and graphs last time we introduced the concept of the limit so let's review that suppose the function f of X is defined for all X near a except possibly they itself the limit of f of X as X approaches a is L means if we can make f of X arbitrarily close to some number L by restricting X to be sufficiently close to but not equal to a so we write it with this notation here we have the f of X and then to the left of it L am and then below it X approaches a the arrow and if that happens to be a number then we'll write that as equals L right so the idea here was that we wanted to see what happened to the function as X approached a we don't really care what happens a we want to know what happens as X approaches a because it's most important at places where F of a is undefined but near it it is defined so let's amplify this a little bit more one important thing to realize is that when we say X approaches a it could approach from two different directions and they may not do the same thing from those two different directions so we have notation to talk about the approach from either side and so we have the idea of the limit of f of X as X approaches a from the right or from above is equal to L so the notation looks just like the limit sign except we have this superscript plus on the a there please understand that that is that indicates that we're coming from the right or from above that is X is approaching a but we only worry about values of X that are bigger than a the other one-sided limit comes from below or from the left so the limit is f of X as X approaches a from the left or from below right we have the notation here we just changed that superscript plus to a minus so this is the limit as X approaches a from below or from the left of f of X is equal to L here we only worry about values of X that are less than a and approaching a there's an important theorem that connects what we did last time with these two concepts the limit as X approaches a of f of X is equal to L if and only if the limit as X approaches a from below of f of X is equal to L and also the limit as X approaches a from above of f of X is equal to L the two-sided limit only exists if the two one-sided limits exist and are equal to each other there's a good example for understanding this let's look at the function X divided by the absolute value of x and we'll take the limit as X approaches 0 from below the limit as X approaches 0 from above and then just the limit as X approaches 0 so let's simplify this function a little bit first think about how the absolute value behaves if X is greater than or equal to 0 the absolute value of x is just X if X is negative the absolute value of X is something you obviously know how to compute you just throw away the minus sign but when you're writing it like this you have to think about how you actually do throw away that minus sign and you accomplish that just by multiplying by negative 1 if X is negative the absolute value of X is minus X and so if I divide X by this representation of the absolute value of x when X is greater than 0 I just have X divided by X that's 1 but if X is negative I have X divided by minus X so that would be minus 1 and then if X is 0 then this function is undefined because 0/0 is not number right so this function that I'm interested in is one if X is positive and negative one if X is negative so think about the graph of that right it's a horizontal line and then there's a jump up to another horizontal line at one it's a horizontal line it's a horizontal line at y equals negative one until we get to zero and then there's a jump to a different horizontal line y equals one and now let's think about the one-sided limits that were looking at here as X approaches one from the left we have only values of X that are less than zero well all of those values are negative one so I think it's pretty clear that this limit has to be negative one and so for X divided by the absolute value of X as X approaches zero from below this limit is negative one on the other side right the function is always 1 and so as X approaches zero from above this limit would be 1 now you may notice that 1 and negative 1 are not the same number so that means the two-sided limit does not exist it cannot be both 1 and negative 1 simultaneously and so it's nothing so we have two one-sided limits that exist but disagree so the two-sided limit does not exist not all limits exist we say a limit exists if and only if it is equal to a single finite number and that didn't happen in the previous case let's look at another couple of examples where the limits don't exist so let's look at the limit as X approaches 3 from below as X approaches 3 from above and then just the limit as X approaches 3 of the square root of 9 minus x squared so that function should look familiar to you it is the top half of the circle of radius 3 and so I'm going to approach 3 from below the values of X that are less than three right we just plug in some values here and it seems like those values - those about the values of X approach three the values of f of X are getting smaller and smaller and smaller right and so this limit is going to be zero from the other side though we have a very big problem the domain of this function is only the closed interval negative 3 to 3 and so if we're trying to approach 3 from above we're approaching 3 from outside of the domain of this function and so we cannot evaluate the function so it just doesn't make sense to talk about this limit at all so this limit does not exist so that means we have a one one sided limit that exists but the other one-sided limit does not exist and so the two-sided limit cannot exist either one more example the limit as X approaches zero of the sine of PI over X this one is rather tricky to think about let's just start by plugging in some values and seeing what they approach all right so we need two approaches zero so let's look at point 1 and negative point 1 we get my plug those in and we get zero point zero zero one point zero one and negative point zero one we get 0 zero point zero zero one and negative zero point zero zero one we get zero so this would suggest that the limit is probably zero and there's something that's not that's not the best way to think about it right there are lots of numbers that we're leaving out so let's take another sequence of numbers that approaches zero let's look at the I'll focus on the positive ones here let's look at the number two divided by twenty one if we plug that into this function that is going to be one and then two over 200 and we plug that into the function we get 1/2 over 2001 we plug that in we could want as well notice that each of those 3 positive numbers is just to the left of the values that I used initially 0.01 the zero point one zero point zero one zero one zero zero one alright so in between every pair that I have in the first row I have another I don't have another number in between that looks like the limit is going to approach one and I can do from the other perspective right I can look at the sequence two over twenty three if I plug that in I get negative one two over two hundred three I get negative one two over two thousand three I get negative one and so that would tell me that this should be negative one this is a really really big problem right what's going on here the sign wants to oscillate but as X approaches zero PI over X gets very very large and so this function is gonna oscillate very rapidly as it approaches as X approaches zero and that's really hard to think about right the limit cannot be zero one and negative one it can be at most one value and it's it's a little bit worse than that right we can make an equally strong argument that this should be every number in the interval negative one to one so this limit does not exist it does not approach a single finite number the example that we just worked is a good example of why plugging in values like that and trying to read off a table is not sufficient if we tried that but we would just plug in zero point ones or 1 0 1 0 1 0 0 1 and we probably conclude that the limit was zero and that's because we didn't really think about looking in between all of those numbers and we can always look in between them there are infinitely many numbers that were not considering and they may do something significantly different than the numbers that we are considering that's a pretty rare event but it happens and so we have to think about techniques that are general enough so that they don't have that problem some limits on the other hand though are pretty easy to do and for the rest of the time today I'll focus on reading limits off of a graph if we've got the graph of the function the limit is usually pretty easy to read after today we'll look at how do we do this without the graph so I've got a rather nasty looking piecewise functions probably one two three four five six pieces at least right but it has a graph the picture of it is not too hard to understand alright so let's think about what the limit as X approaches 2 of f of X means right we come over towards x equals 2 and we look on either side of it and we see that there are two radically different kinds of curves but they're both approaching the same value as an X approaches two the function is approaching three from both sides so the limit as X approaches 2 of f of X would be 3 f of 2 is what happens at x equals 2 and we just follow the vertical line up there and it's at 3 and so here the limit as X approaches 2 of f of X was equal to f of 2 if we approach 4 there's nothing different as we approach four we have two different lines that are approaching each other and it looks like the Y values of those two parts of the curve they're approaching each other they're approaching two so this limit would be two what happens at 4 right when X is equal to 4 the part that we had just been looking at how to hold where the hole is not fill in there is a dot down one and so f of 4 is equal to one that is completely separate from the limit as X approaches four that limits - so limits can equal the function values you get when you plug in or they might not here is a one-sided limit what happens as X approaches one from below so we come to the left of x equals one and we follow the curve up here it's a line that line is approaching that open circle at 1/3 so the limit would be three right the function is approaching 3 as X approaches one from below on the other side of 1 we have a different piece we have this arc bending down the limit as X approaches 1 from above f of X is 2 it's what happens when you approach on the other side of it you know on that different piece and you approach a different open circle well these disagree right on one side we the function approaches 3 on the other side the function approaches 2 those are different and so the limit as X approaches 1 cannot be anything it can't be both 3 & 2 simultaneously now what about F of 1 well we look up above and we don't find anything we find two empty holes we find nothing filled in so f of 1 is undefined typically we're trying to say function values are undefined and limit values do not exist I'm not gonna count off if you mix those up but that's typically the way we save it now let's think about the limit as X approaches negative 1 from below this is doing something a little bit different we have this dashed vertical line to indicate that we have a vertical asymptote and so this is approaching this is increasing as it approaches that asymptote so this would say that this goes to infinity we will talk a lot more about this next time on the other side we don't really have it in trouble right if the limit as X approaches negative 1 from above we just have lennier piece that we were looking at the other side of earlier so that decreases and it goes down to one and so the limit as X approaches negative one from above is one from below it's infinity infinity is not a number and it's certainly not equal to one so the limit as X approaches negative 1 of f of X does not exist F of negative 1 does exist so here we have a vertical asymptote but the function is defined there and it's a vertical asymptote from one side only we'll talk more about this later