Calculus 1 Lecture 3.5: Limits of Functions at Infinity

3.5 we're going to talk about limits at Infinity I'll start off with just a little bit of a recall here do you [Music] remember if the limit as X approach a number of f ofx equal plus or minus infinity do you remember that that gave us an ASM toote so basically when you could not cross out the problem on your Limit you had an ASM toote when you could cross it out it's called a removable discontinuity is called a hole this right here will give you an ASM toote at the value a and we had

two cases really we have this case well we get went to the same Infinity positive Infinity on both sides and we said the limit existed the limit was positive infin there or we had this Case where one went up another went down we said the limit didn't exist there one's going to positive Infinity as we go from the left one's going to NE as we go from the right and we had a way to actually determine this if you remember uh how did we determine where we had holes or ASM tootes s test would do

this but how do we find out where all this stuff happens let's pretend we had a rational function Rational functions have denominators how do you find these things denominator it's a and if you can't okay let me give you an example will this have some discontinuities to it absolutely where at zero 1 three which ones zero is zero going to be a discontinuity what do you think zero no zero is fine Z over numbers okay that's okay uh where are discontinuities going To exist 3 one why good because if a denominator equals zero you have

a problem right that's how you find discontinuities that's what I told you a domain from a long time ago so what do we do to find discontinuities denominator equals z that's how we do it we we said denominator equals z we solve it down uh we're going to get either holes or asmp tootes do you remember When we get holes and when we get ASM tootes okay maybe we'll refresh your memory on that next time so if you'll recall when we take a limit of a function as we approach some number and it goes to

Infinity we're going to have an ASM toote we're going to have either an ASM toote where the limit exists as they both go to positive Infinity I didn't draw the case where they both go to negative infinity or where one goes a positive one goes a Negative and then we find that with the sign analysis test now there were two cases where discontinuities existed we had holes and we had asmp tootes do you remember those the holes in the ASM tootes the holes was what was called a removable discontinuity what it meant was that only

a single point is missing from that function one little single point that was what defined a removable Discontinuity and it exists when the numerator and the denominator equals Zer at the same time why was that the case well if both the numerator and denominator equal zero at the same point you're going to have a factor of x minus that value that you could cross out remember Crossing out those those problem areas that was a removable discontinuity now or a hole if we don't have that if the numerator equals a number and the denominator Equals zero

and you cannot cross it out and you can't simplify it it's not removable that means we have an ASM toote so case two is where we had the ASM toote A vertical ASM toote A vertical asmt this is when you can't cancel out I know you love to use that word cancel out right where you can't cancel out the discontinuity or you can't remove it Let's do an example here real quick okay Let's do an example can you tell me where discontinuities exist so any discontinuities how about x equals z is that a discontinuity or

not is it okay to have zero on top of a fraction sure that's zero that's a point uh give me one disc continuity one one would work sure because if I plug in one I'm going when to get zero in the denominator that's a discontinuity you follow me so x = 1 Will be a spot at which we will have a discontinuity give me another one someone left hand side of the room what do we have3 would be another one sure now let me ask you a question about x = 1 x = -3 are

they holes are they ASM tootes is one a hole is one an ASM toote what do you think and the question is can you cross out where the discontinuity happen so basically can you cross out the x -1 can you cross out the X+ 3 so are those holes or ASM tootes those are both ASM tootes both of them now stop for a second just watch what if I had done this please which one would be the ASM toote in this case that would be the ASM toote here and that would be a removable discontinuity

or in other words a hole do you follow that okay good so I'm going to change it back to this this is an Asm toote And that's an ASM toote if I have just x x + 3 and x -1 those are vertical ASM tootes do you recall how to find out what the limit is around those do you remember that yeah signalysis is how you do it you'd put the3 and you know it's going to be ASM toote because we just talked about that it's going to be ASM toote there and you put POS

one and you know it's going to be an ASM Toote and then all you have to do is plug in some numbers for those intervals because look it look if it's an ASM toote look at the board here real quick you know it's going to be one of these cases right it's either going to go this that or be opposite so it's going to be one of those all you have to do is plug in numbers if it's positive it's going positive if it's negative it's going negative same thing in here positive positive negative negative

same thing Here positive or negative now now there's one thing you have to be careful of one very important thing for us right now do you see that the numerator will equal zero at zero that Point's not give me an ASM toote but it will it will change the the sign of your fraction so you're going to put a little marker in here at zero and the only reason why you're going to do it is because a number in this region could potentially be different than a Number in this region concerning the sign does that

make sense to you so that's a separator because if you just plug in zero you're going to get zero right that doesn't tell you positive or negative that could be a case where you're switching from positive to negative so you have to put that plug in a number here and plug in a number here and that'll tell you so it's like you have just three intervals but that's a separating marker which says I Can't plug in just one number here I need to plug in two different numbers so you only have two ASM tootes there's

not three ASM tootes but it's like yes like you're testing four intervals okay you okay with this why don't we see what happens uh why don't you plug in ne4 somewh how about this uh U2 rows do -4 U2 rows do1 U2 rows do 0.5 and you two rows do two okay can you tell me that have you already done the two when you have them let me know it's Negative so you plug you plug it into here right you got negative I don't care what the value negative do that uh these two rows have

you plugged in negative one negative really negative positive positive how about5 positive POS negative how about two here's my Question will the limit exist will the limit exist as we approach3 from the left we're going down from the right we're going up does it exist heck no how about at one nope from the left we're going down from the right we're going up if they had been like this both going up both going down sure how would this question trick question uh does the limit exist at zero absolutely that's a point oh yeah there's no

as tote there right that's The point liit does exist at zero trick job take it job feel okay with this so far could you do the same thing with something like this the same thing with something like that can you tell me where discontinuities exist on this problem where would they exist on this problem do you need to distribute all this out to tell tell that the only place where discontinuities are going to exist is Where the denominator equals zero true so where the denominator equals zero is where this little piece equals zero you follow

where is that going to equal zero uhhuh so these are my discontinuities would you say that those are holes or ASM tootes as explain why they're asmp tootes because they can't be deleted okay they can't be removed that's exactly right can't be deleted is What she said but absolutely if even if you factored that could you cross it out at all definitely not so these are non- removable discontinuities those are ASM tootes so how could you find out where those ASM totes go well this says really what we're concerned about is around the one do

you see what I'm talking about so from our our chart here we've got two discontinuities we got1 we've got one 1 and 1 and what we're going to do is check where these ASM tootes go now look up here at the board here real quick normally we would check all the asmp tootes follow but which one do I really care about do I care about the negative one why not limit says one so I don't care about that I care about this sure tell me one more point that I absolutely must have up here right

now that's zero that zero could be a change of sign so where the Numerator equals zero that could also be a place where you want to at least separate your interval it's not going to be an ASM toote but you definitely want to plug in just to be safe a number between zero and one that's what you want to do so try that people on the left hand side want to try 05 right hand side wants to try two for me see what happens around that shouldn't really be able to do it without a calculator

honestly U because When you think about it this is going to be a positive number right positive doesn't matter that's positive positive over positive is positive negative really number5 is a positive number you didn't check negative did you cuz we should be checking between this little interval that's why we had that okay5 is positive this is positive positive or positive positive ah ah we this first time I think we've had one of these does the Limit exist yes yes it does this limit as T approaches one of that nasty looking function is what can you

tell me positive or negative so our sign analysis is important our analysi how do you say that analysis analyses oh that's better our s analyses important for you are they important yeah you got to know how to do those so when you can Cross cross out it's very easy those are holes no problem when you can't cross them out you have to have some way to determine where they're going and the way you do that is with a sign analysis just like that don't worry about all the ASM tootes if I don't ask them to

for you worry about just where the limit says if I'm asking for the all the discontinuities I'm just asking for the points right this was a little extra for us in that case I just want to see what Happened this is the application for a limit I feel okay with this so far all right good now the next question we have to answer is what happens with the limit if x doesn't actually go to a number so for instance X goes to one here right or t in this case a variable goes to to one

what if we say I don't want to go to one I don't want to go to two I want to go to Infinity what happens to our function I don't want to go to3 I don't want to I want to go to negative Infinity what happens to our function that's what this next part is all about so the question is what happens in a limit or to a function as X approaches positive Infinity as far as we can we can go or negative Infinity basically X increases forever or decreases forever what happens to our function

let's take a look at two Examples that are going to give us some insight as to this um you know what let's make a table just to kind of get a a picture of what's going on we'll have X and F ofx we'll start very easily we start at 1 10 100 and then a million let's do it for this one what would happen please if xal to one what would we get One how about if x equal 10 what would we get 0.1 very good or 1110 yeah clearly right one over whatever that number

is how about 100 01 how about a million six zeros right five Z five zeros yeah five zeros what if x equal a trillion would it get bigger or smaller smaller small what's it going To is it ever going to reach zero but think about this you're dividing a constant number by Infinity right like when you're a little kid and you go you're wrong well you're more wrong well you're wrong plus 10 well you're wrong times infinity right well if you did divided by Infinity you'd have like you'd have zero you keep dividing by a

bigger and bigger number you're going to get zero right one divided by a billion is very Close to zero one divided by a zillion not say number number is very close to zero but you keep getting bigger and bigger this quantity is closer and closer to Z you with me on that zero let's talk what negative Infinity what happens as you go closer to negative Infinity well that would be I'm just going to change this that would be negative negative negative negative true negative negative negative negative Is it still approaching zero yes as we go

to nak Infinity we're also approaching zero what this says is that if f ofx function gets really really close to a number as we approach Infinity then the limit exists so we can take limits as X approaches one of these Infinities if x approaches an actual number remember that idea of really close to that was the idea of that limit if FX Gets really close to a number as X approaches infinity plus or minus whichever one then the limit exists okay well if the limit exist what's it stand for what's it mean think about this

let's say that you start taking your function and you start going forever in that direction it's almost like a karate move I take karate I know but it kind of feels like it whatever the limit move so if you by the way if You ever try to do the math karate you're going to get you're going to beat up never do math karate so you go this way forever and ever and ever and your functions going into the same number is it ever going to get to that number no let get very very very very

very close right what idea do we have that represents that but not a vertical one a vertical one says you're going to get really close to that right but never reach it That's the same idea we have going this way really close to that but then we don't have this idea we only have one because otherwise we don't have a function but we have this idea we're going to get close to a number or close to a number close to a number or close to a number that's a horizontal ASM toote so now we have

two types of ASM tootes vertical happens when you have discontinuities that are not removable shake your head nod your head if you're Okay with that horizontal happens when you take a limit of your function to infinity and that's the idea horizontal asmt by the way the same rules apply to these limits we're about to do uh the rules for limits as X approaches negative and positive Infinity are the same as before so the same rules apply everything holds the same which means one thing for Us consider this what if we take any any 1/x n

power do you recall from limits that I can take the limit of the function and then raise it to the power do you remember that that we could pull out the exponent that was one of of our Prime rules that helped us a lot well notice how 1 to the N is always just going to be one right so then this is true that's a true statement that's 1/ n I'm sorry 1 to the N that's one this is X to the N that's x to the N are you guys okay with that statement well

tell me something then how much is the limit of 1x as we approach Infinity what's 0 to the N power so that what this says in English is that any function where you're dividing a constant one because you can pull a constant out any constant can be pulled out of that limit any constant by some variable that's being raised or as X approaches to that that Infinity is Going to end up being zero it's going to have a horizontal ASM toote at zero 0 to the n or zero does it make a difference if we're

talking about negative Infinity let's consider that let's go to negative infinity negative Infinity that means is this still true for negative Infinity absolutely is this still true is 1 /x is X approaches negative Infinity is that still zero zero to that holds true for Both of them what this says is that one a constant over any variable that's being taken to Infinity it's going to give you a horizontal ASM toote how people feel okay with that that idea no matter what the power is squar Cube doesn't matter because you're always taking that 1 /x to

that power 1/x as X approaches Infinity either positive or negative it's going to be zero zero any power is still zero interesting thing horizontal ASM tootes in each case At zero now before we get into some other computational stuff let me show you one one more idea right now we've done a lot of the so far so there no not really a whole lot of examples we're going to change that in just a minute but I got to talk about one more I want you to think of polinomial let's talk about the limit of a

polinomial As X approaches positive or negative Infinity think about what you know what you know about pols would you agree that pols are in one of these four one two one of these four cases pols are either like this like this like this or like that true because think about x s x squ like this does it ever go to an actual number As we go this way does it ever go down or up and go to no pols don't do that pols always have a tail somewhere uh an xq would go this way to

that way right X Cub goes this way to this way X2 goes this way X2 goes this way they never actually go to a actual number they're all either going to positive or negative Infinity so as we approach for any polinomial positive or negative Infinity on the x- axis the function itself approaches positive or negative Infinity On the Y AIS let me explain that one more time because I think I lost something as you travel this way on the X your function is doing one of two things going this way forever or going this way

forever you follow me on that one that's what I just said okay if you're going this way on the x-axis your function is doing one of two things going this way forever or going this way forever that's what this says the limit of a polinomial as X Approaches negative infinity or positive Infinity goes to positive or negative Infinity itself I'll give you some examples of this by the way you're going to notice that I always use positive and negative Infinity that's something I do because I don't like to get confused I I if I say

positive I I write the positive that's just way I do it do you have to do it that way no not Necessarily but it helps helps me helps me let's let's consider this okay now Infinity is not something we can actually plug in but something think about in fact this one guy actually went crazy thinking about the sizes of infinity because there's different sizes of infinity which is pretty interesting itself but so thinking on Infinity is kind of weird to think about uh but we can do it think about this take a big positive Number

plug in X Cub you know the shape of X Cub right it's a polinomial it's going to be like this take a positive number and plug it in do you get a positive or negative take a bigger number is it still positive take a bigger number is it still so POS in fact it's getting bigger and bigger this is going to go to positive [Music] Infinity the bigger the positive number I plug in the bigger the positive number I get out let's try this one negative Infinity take a negative number plug it in what do

you get take a smaller negative number even more negative plug it in do you still get Negative do you see that this one's actually going to negative Infinity further we go l on the number line the further we go down on the Y AIS and that's what that says the Further you go left the further your function drops that's what that's saying does that make sense to you how about this one x s take a positive number plug it in X squ where are you going up or down up positive Infinity so the bigger the

positive number the bigger the positive number try that one though take a negative number plug it in what do you get ah a positive cuz x s takes that number and makes a positive so even Though we're going to the left our function is going up that's why x s looks like that right it's going to go up on both sides this is also positive Infinity show it hands how people feel okay with our our limit ideas so far good all right all right by the way this kind of an interesting little note also did

you know did you know that the behavior of the polinomial itself will Follow the behavior of the leading term did you know that I'll explain that in a second sorry I'll say this way limits of a polinomial will follow the behavior of the highest highest power of term polinomial nothing else Pol let me give you one quick example because I think you'll really handle this pretty Well let's say that I want to talk about the limit as X approaches negative Infinity of -3x Cub - 2x^2 - x + 9 now here's what you can't do

what you can't do is go oh let's see uh negative Infinity cubed Infinity * negative that's positive Infinity minus A positive Infinity that's oh man I if you start doing stuff like that subtracting Infinities that is going to make you go crazy what you need to understand is that all this subtracting and adding the infinities are meaningless the behavior of this polinomial is going to follow the term that has the largest power power the term that has largest power is usually in front it doesn't have to be but it's usually in front of your polom

it's this One here's what this little Corr says what this theorem says it says this is going to be equal to the Limit as X approaches of just the -3x Cub don't forget the negative that's important but the -3x CU you know this intuitively in fact when you think about it look at that the power is three don't you automatically know it's going to beur for sure it's going to be doing this a power four would look like this five 6 7 Eight you know that right now negatives would be opposite so negative X cubed

negative of that no n Negative X Cub Negative X 4th X 5th it's going to do that right you know thatly it follows that leading term term all the rest of this is garbage as far as limit is concerned really when you're talking about Infinity now normal limits no it's not garbage but for Infinity that's the one that matters so consider the the leading term that -3x 3r take We're going to negative Infinity mind you take and do this negative means negative right means a negative number take a negative number plug it in here what

do you get out and then multiply it by3 what do you get a positive or negative that's positive so as we're approaching negative Infinity that approaches positive Infinity it's polinomial will go to one Infinity let me do this one more time so you really catch this on You're talking about going to really negative numbers right really negative plug in negative numbers to that you're going to get out negative numbers right but then you're multiply it by3 which makes them positive numbers that's positive Infinity think about Cub if you think about XB it looks like this

doesn't it about like that let's see move my arm bit there it looks about like That doing Dance Dance Revolution or something uh it it's this as you go left it goes higher doesn't it as you go right it goes lower that's what this is saying that's what that's saying how many people feel okay with that one I'm not trying to show limits here I'm just ask okay we're going to start using these ideas to actually compute some limits this is where kind of the fun comes in we're back to Computing limits are they Hard

not really but you need to know not tricks but some mathematical manipulations to do them correctly there's going to be a lot of rationalization a lot of that there's going to be a lot of dividing by things there's a lot of that not hard though you just got to see it a few times okay what about that one let's see if we can do that what About that uh where are we going positive Infinity really big numbers so plug in posit what's 5 * positive infinity infinity positive right minus 2 Infinity okay what's 3 *

infinity plus 9 I got you infinity plus one what Infinity over infinity one yes no No you can't do that do you know how big Infinity is me neither so how you supposed to divide and give one you can't you can't do that but maybe we can manipulate this in such a way that this stuff is possible now here's the idea did you know that you can multiply both the top and the bottom of a fraction by anything that you want sure and if you multiply both the top and the bottom of the fraction by

Anything you want it goes to both of those terms true what that says is this what if I divided every term by X could I do it mathematically is that legal sure mathematically that's fine 5X - 2 3x + 9 let's divide all those terms by X I'll explain where I'm getting the X from in Just a minute now some of you can probably see it I'll explain why in a bit you okay with that so far you sure okay now simplify what happens here same thing happens here right do they cancel out or cross

out here so I get the limit this becomes 5 - 2x 3 + 9x but I show a hands how many people Can follow that algebra down now watch let's take X to Infinity if I take X to Infinity what happens to the five nothing nothing there's no X it doesn't change at all the five is five what happens to the three nothing what happens to 2x really small how small zero zero remember the definition of our limit our limit said as we take X approaches Infinity a constant over X is the first First thing

I gave you today really the First new thing said a constant overx to any power goes to what zero that was important we needed that this becomes how much folks everybody how much does this become Z constant over infinity whether it's positive or netive doesn't matter it's going to zero so what this says is this limit notice how I'm going to stop writing limit now I've wrot limit till here 5 - 0 over 3 + 0 that's 5/3 not one 5/3 Now is it coincidental is it coincidental that uh by the way this works you

do 1X over 1X do you guys see what I'm talking about there what I really did mathematically was this over that do you see it one overx over one overx and that distributed and that basically divided every term by X that's the mathematical way youd show that look at the first look at the first expression the limit of 5x - 2 3x + 9 What's the Le leading term here 5x is a leading term what's the leading term here what are the coefficients do you see that when I divide excuse me when I divide by

that X I'm going to get 5/3 everything else is going to go to zero so the lead those are polinomial the leading term is all that really matters for us for that now do you have to show it yeah I'd like you to show it for right now okay as you're going on you know later math classes no Problem but but show me that where you getting X from is not that the common X here you're actually only looking at the denominator and what you're going to do is divide every term by the largest power

of X in the denominator you should probably write that down now why not the numerator well think about it if your if your exponents don't exactly match up and you start dividing by the largest power in the numerator You could have a lot of undefined things does that make sense for instance if you had this uh X to 4/ X cub and you divide it by x to the 4 You' have other stuff here but if you divide by X 4th that's undefined that's one /x that's a bad thing so it's always by the largest

power in the denominator that way you won't be undefined you might be going to Infinity but you won't be undefined does that make sense to you okay another way uh people like to Say this I'm not going to write down it's kind of it's a shortcut um but if you're your powers are equal to each other you're going to the coefficient over the coefficient that's what we have here but you can see that that's going to work right x s X2 X Cub no problem if your power is larger on the numerator than the denominator

well you divide by this power this is still going to have an X up there somewhere right that's Infinity Either positive or negative Infinity if this one is larger than I'm s smaller than this one if this one's larger well then you're going to be going to zero cuz you divide by this one one no X's here x here you're going to zero that's one way you can look at that as well so if the top still has an of course let's try a couple more you're going to you're going to start seeing this very

easily it's not going to be Something super duper hard what's going to be super duper hard is we going to combine this with some other ideas and little while actually might super duper hard then just super hard not even the duper don't worry about the duper come on that was that was funny that was among others if you were listening to what I was talking about you can probably tell me where this limit is going right off the Bat not one third if the powers matched up it would where's the power the biggest that means

what we're going to be doing here to show you work you're going to be dividing everything not by x2m but by x to the thir power x to the thir power you follow me on that so this would look just like this you'd have 5 x^2 over x 3r - 4x over X 3r all over everything get divided Everything 15 x 3r over x 3r minus 3 x 3r if we do just a little of simplification well look what happens do you see how this power it's still still going to be 5x and this is

going to be 4 x² you see the x s this is going to be 15 sure but this is going to be 3x to the 3 we had that that the principles of beginning class for a reason it said any constant over any power of X as X approaches any Infinity is going where folks where Z zero so where's this going Z zero where's this going where's this going oh come on come on everyone got to play along where's this going 0 0 15 0 so this after you stop running the limit would be 0

- 0 over 15 - 0 that's where these things are going it's a constant over X as X approaches infinity negative Infinity but still an Infinity 0 0 0 0 how much is that going To be0 that's zero that says a horizontal ASM toote that's what this is by the way a horizontal ASM toote as you're going to the left are you following me on that now if you did this if you reciprocated this and you divided everything by X squ do you see how you still have an x on the numerator that would be

saying you're going to positive or negative Infinity depending on what U what x is approaching let's do one More and then next time we'll start building on this coniss two more two more we get we get three minutes now there again some of you who kind of understand this this concept right now should probably be able to tell me where this is going to approach or at least have a guess where is going to approach Okay okay o Battle Royale out well you might want to do the work to make sure but you see it's

not going to go to a constant do you see it's not going to go to zero what are we going to divide by X Cub x s or X the largest power in the denominator so show your work don't get lazy on that don't take a guess cuz some of you were guessing right now don't take a guess show your work over X Overx overx all over that divide everything by the largest power in the denominator not the numerator I give you undefined spots do you see that if I divide it by X Cub I

get it's a whole bunch of undefined stuff can't have that can't do that are you all right with that one so far simplify them you're going to get 7 x^2 - 2x + 1X all over 9 /x - 2 follow me on that now let's kind of think carefully about what happens here what happens where's that go that goes to positive Infinity does this stuff matter no that goes to positive Infinity agreed is x s this is zero but that is a -2 where's that going negative Infinity in fact if you really wanted me to

show it to you this way I could could you factor out a negative and get 2 - 9x move the negative up top bam got it do you see it that L you you understand that went to positive Infinity little unorthodox here but you see this goes to zero netive two doesn't change positive divided by negative is a Infinity divid two still Infinity Okay now mathematically oh not that doesn't do that that mathematically the way that you could show this is the way I showed you if you factor out the negative it's 2 - 9x

right if you take that negative and move to the top of your fraction which is legal to do it goes there that becomes 2 - 9 x that says this goes to Infinity time a Negative that's negative Infinity divided by a positive now that's negative Infinity that's what that le as well we show yeah by the way did anything but the highest powers of X have anything to do with our problem here this one and that one was really what what caused this to be the way it was that uh one last thing I'll leave

you with just a idea because I want you to look at some of your homework Tonight but I don't want you to be completely afraid of it just a little afid that looks pretty nasty but tell me something that I can do with exponents what can you do X not yet I can move them outside is a root a type of exponent then look you could take a cube root of the [Music] limit ah now ignore the cube for a Second just for a second though can you do this limit yeah explain to me how

you would do this limit what would you you divide by here x which one x s Hey where's this limit going to go from on the inside do you see the 2/3 2/3 and then take a cube root of that so our answer here after you show the work would be a cube root of 2/3 the work you show is is this you Show a cube root 2x^2 X2 - 32 all over that's what you show but that's the answer you're get how many people understood today if you okay with it right side people are

you guys with this so far yeah okay okay so we're talking about how to do limits as X approaches Infinity now the last example I gave you I think I Gave you that one didn't I I said that when you have a cube root it is a power that means you can pull it outside of our limit and says well let's take the cube root of our limit itself now when you do that that's kind of nice right because we know how to take these limits what we normally do is divide everything by the largest

power in the denominator that's X squ you would get two that's going to be zero three that's going to be zero you're going to get 2/3 you Follow don't forget about the cube root but your answer is 2/3 with a cube root around it I think that's what I gave you last time were you okay with that one yeah now I I showed you that one again because I want you to consider the problem that's just below it can I do the same thing here for instance can I pull out the square root around the

whole limit no no because that square root is not around This whole function it's just on the numerator so oh my gosh well what do we do what do we do well let's let's try to stick with the the normal operation of this when you're taking X to Infinity we can't just go well that's Infinity s okay that's still infinity plus 2 Infinity Roots Infinity over Infinity uh one we can't do that but what we did say was maybe we divide by the largest power of X in theat denominator so we're going to look Down

there what's the largest power of X in the denominator x to the first power so let's divide the numerator and the denominator by X are you okay with that so far well now here's the issue and some of you might see this issue already can I just take this x inside that square root I can't unless I have a square root remember you can only combine combine Things if you have the same exact type of root follow because it's it's an exponent you can only put things together if you have the same type of exponent

well well that's a problem does any know how to change X into something I can take into that square root let's think about it for a second let's think about that x what do you think but if I do x s I have to change those to x S right is that a problem that would be zero and zero that would be undefined so I can't just arbitrarily change X squ but you're on the right track 2 over two is still One X well I did did that so x s down here we definitely we

can't have an X squ I need this to still be a value of x make it a value of x s is a great idea x * X is x 4 that's too much Ian say what now rooot of x * root of x is still X you're close oh you're so close you're so close put that together you want something that equals x true how do you make this equal x do that and they are equal is that equal okay no no but almost that's our idea though I'm going to fix this in a

second so here's here's the plan well I know that I need this do you see why I need that that is still well pretty much X almost X and that's going to allow me to take that Into that square root do you follow me that's important but the question is is X = to theun of X2 and the answer is no not really because if you think about it why don't you take like a negative number5 is5 equal to theare < TK of5 S no it's not because as soon as you square it becomes positive

right so this is not necessarily true what is true is this that's true so if we have to have this there's Something that's wrong in our problem right now this is right what is wrong do you see it you don't see it um let me go through the process one more time you knew maybe if I draw in purple it'll make more sense you knew if you you had to do that right but you knew that this wouldn't work so to try to make it equivalent you did This that would go into into the square

root however this fails now because this is not equal that's not equal notice you you're trying to divide everything by the same thing right the same thing so if you have divided by the same thing these are not the same thing this is the same thing that's true this is true yes this is not true not necessarily true so are the absolute value important This is almost true is this true well no almost this is true you ready to keep going do you understand that that idea right now you sure okay so when we take

the square root of x squ we can't just have X you have to have the absolute value of x and I'll show you why it's important in maybe 10 minutes for right now let's continue working on this problem when you have a square root over a Square root that says you can take a square root of the whole entire fraction that's x^2 X2 + 2x^2 are you okay on where the x^2 X2 and 2x squ are coming from that's the whole Sol reason why sole reason why you did that right just to make that in

there over 3x over absolute value of x - 6 over the absolute value of x well that's a limit X goes to positive Infinity you're going to have the square otk of 1 + 2x^2 all oh okay okay now we got to deal with that I need your mind to think back to what it means to be an absolute value positive values that absolute value is actually a peie wise function the peie wise function says if x is positive you leave it alone if x is negative you change the sign does that make sense to

you if x is positive Positive you leave it alone if x is negative you change the sign and that was the whole idea that's the piecewise definition of absolute value did you remember that I think we did here actually well this is how you apply that we are going to have the 3x over - 6 over now you get to change your absolute value of x depending on where you are going whether you're going to positive infinity or negative Infinity where are we going to positive Numbers or negative numbers we're going to positive positive numbers

say that absolute value of x equals x positive numbers positive numbers say that absolute value of x equals x are you with me on this absolute value of x now you don't need the absolute value you just know that since I'm going to positive numbers that can become the X it might seem trivial like well if it's just going to change to X anyway why do we need the absolute value trust me you Need them I'll show you why in a couple minutes okay now like five minutes ra going to feel okay with that so

far you see now that we have that those X's do match up those X's are gone you get a limit to positive Infinity 1 + 2 x^2 all over 3 - 6X tell me some nice things that happen since X is going to positive Infinity tell me what happens where does This go not zero one this the one oh one the one goes to one it's constant do constants go anywhere no where's this go where's the fraction go have we lost it all on G you're scaring me people you're scaring me here limits of constants

are constants limits of this is not undefined where's X going X is going to Infinity where's this fraction going zero constant over Infinity is zero divide a constant by Infinity how much you going to get zero zero think about that come on where's three going three okay where's this going zero don't forget what we did last time this says you're going to do 1 + 0 over 3 - 0 tell me my final answer please what is that are you okay on getting the zero and the zero I think we talked about that last time

a constant over any power of X as X Approaches Infinity is going to be zero because this number is getting huge and this number is getting huge and those numbers are staying the same that means you're going to have a number over huge a number over huge is zero 13 not the square root of one okay one3 show hands how many people feel okay with that now I want you to consider this why this is important think about it you can change your problem if you Want to I'm not going to do the whole example

over again I'm not going to give you two examples I'm going to show you with this let's say that that is now negative that means that's negative is this still possible absolutely is this still true absolutely is that still okay yes now at this point something different would happen at this point from here to Here you see if we're going to Infinity that means that for our absolute value of x where are we going are we we're not going here anymore we're not dealing with positive numbers we're dealing with negative numbers that means instead of

absolute value of x equaling X absolute value of x would equal x that means that this and that would be there do you see the change this would be okay squ right square is going to be positive no matter What that's where our definition came from but this would be different what's 3xx - 6 over well that's a plus that's not really going to affect us because that's being plus Z but this definitely affects us doesn't it that's -3 + -3 + 0 that's 1/3 not POS 1/3 what that says is this is interesting the

first time we've had this because if you've been thinking about it you might have said well why do we even take positive infinity and negative Infinity why don't we just do One infinity and consider it a horizontal ASM of the whole way well it doesn't work look at that our first example was positive Infinity right that's that way that was at a horizontal remember what these are a horizontal ASM toote that's a horizontal asot at 1/3 if you go to the left negative Infinity is it at 1/3 no it's a negative 1/3 so our horizontal

asmos can be different as it go to positive infinity and negative Infinity that proves it That's different right there how people understand that cool this row you guys right with that all right hopefully that madees sense to you we're going to do um one more example maybe two just to start it off let's try that uh so you something kind of cool interesting that you can do so we give it a try looks awesome to me I don't know About you looks awesome to me let's try ready uh what's infinity to the fourth power Infinity

okay cool plus two and the S root of that and then minus so Infinity how much is infinity squared zero he done I love that easy problems yeah oh course maybe coincidentally but I really doubt it we can't do that you can't just plug in Infinities whenever you get a case where You're like Infinity minus infinity or Infinity over infinity something like that you know something bad's happening all right yeah you're not doing something right so you can't ever do that you get infinity over infinity please for heaven's sakes don't put one because that that

would have been one right uh we can't do that what we do have to do is manipulate the problem somehow now in this case you have a Root you have a root minus something what's one thing we like to do with roots minus something when we can't work on them any other way we usually take to try to rationalize some right now this doesn't have a denominator so make one put it over what put it over one get yourself a denominator then you're going to try to rationalize conjugate conjugate very good conjugate if you want

to be fancy about It is it plus two or minus two there by the way good so this sign doesn't change this sign does change and it's over exactly the same thing are you okay with that so far yes no well the denominator has nothing to distribute which is really nice we don't have to do any work up there the numerator just distribute it carefully you know with conjugates your middle term should be bye-bye right your middle Term should be gone we know also that it works this way because a square root time a square

root gives you the radican that's inside that's nice so our limit to positive Infinity becomes something over the < TK of x 4 + 2 + x^2 the numerator becomes oh what's the numerator become distribute right do you see that on your distribution Because you should have those x 4 + 2 we're going to have plus and minus the same exact thing and then we're going to get what's the last thing you see the- x 4 two don't forget about the exponent if you make the exponent incorrect if you make the exponent incorrect it's not

going to come out right so you got to really be careful about your distribution don't forget to distribute that last part you got me what happens here that's Nice yeah oh that's great I love that two square root of x 4 + 2 + x^2 so far so good okay all right to zero can we do that legally can we just say two over in well I don't know especially if this is a especially if you did this especially if you did that that Would be a big problem do you see that that would be

an issue so what we're going to do is make sure about it you're going to start dividing by the largest power in the denominator again just like you normally would so what's the largest power in the especially if I gave you this one okay what if that was 2x that would be a big problem this one yeah Scott's right you probably could look at this and go that's two that's infinity plus that's Infinity There's no way that that's not going to be Infinity you got me this is going to go to zero he's right but

to check your work especially if you have something like this that might not go to Infinity that definitely wouldn't go to Infinity okay that wouldn't so to check your work to show that what you're going to do is go all right well let's divide everything by the largest power in in the denominator now don't trick yourself here the largest power in the Denominator is not X 4th it's not X 4th what is it yeah the way you can think about it cover up the rest of it look at the largest term what's the s root

of x 4th it's X squar that's your largest power it's the square root of that because it's inside a square root so divide everything by that I'm making you do this because I want you to see one more time this this setup notice we're divided by X2 if I divide this by X squ is that going to work here is this okay no can I take that inside of that square root okay so think back to what we did this is fine right this is fine that's that's great that's what we like what does this

do what now oh okay we got stuck uh oh X of x to 4th would be great did you say that already yeah I was thinking I was my mind was wandering thinking about The Simpsons um so well since this is x s and this is X2 we need to make this equal to X2 but it also has to have a square root around it you follow you can't just do this cuz that's X so change the power is that true now some of you are going to ask me well wait a second Mr Leonard

don't you have to have some absolute value here let me ask you this question is this always true that actually is always true no Matter what no absolute value needed because you have all positive all positive it's always going to be the same does that make sense so we don't even need that for this case so here we're going to have just wanted you to see that one time 2x^2 all over we're going to have the square root if you think about it this is going to be one plus 2 x 4 do you see

the 1 + 2 x 4th yes no yeah we got the 1 x 4 x 4 plus the 2 x 4th + 1 show it hands how many people feel okay getting down to that far oh that's not many let's have some questions if you guys are are not so so okay on this are you okay on rationalizing yeah are you okay on Crossing those things out that should be easy that's the fun part are you okay on dividing by X squ because that is the Largest power in your denominator you okay on doing the

square root of x 4th cuz that's still x s then we take it inside we have x 4 x 4 that's one 2 x 4 2 x 4 X2 X2 that's 1 where does this go ladies and gentlemen when we're taking it to Infinity where does that go is it okay to have zero on the top of the fraction where's this go where's this go where's this go would you agree that this denominator goes to Two so this equals 0 over two or zero you're going to notice in this case if I change that to

a negative nothing about this changes I don't have any absolute value nothing changes that says the horizontal ASM toote in both directions is at zero that's what it's going to be you follow me on that one would you like to see one more case yeah so would that be all positive exponents or even Exponents would that be the case we just start with the even one there that was I'm sorry was oh I don't know it really depends on what you have I can't say a coverall situation for every time I don't know should we

change the problem just a little bit let's change the problem just a little bit all right so what I'm going to do instead of doing a whole completely new one do you have any Questions on this we're going to see how it changes if I do this little bit to it that okay if that changes our problem at all let's see the things that change let's go through our problem I'll try to use a different color pen so you see the differences can you still rationalize what's the only thing that changes about our rationalization that's

going to be a X2 and x s are you what that so far okay What changes here ladies and gentlemen when I distribute what changes just an X squ you see it' be kind of silly to do the whole example again right it's pretty much the same thing that would be an X squ and this would be an x s you still okay well what happens here then is this is an x s and that's an x s true okay so okay well well hang on a second Then well if that's the case are we

still going to be dividing by X2 down here yeah because that's still the largest power that's that's not that actually if you looked at at would be x square of x square is X so that wouldn't be the larg power we still divide by x^2 we still divide by X2 but now I have an X2 right there that 2x2 that's right there this would look almost the same except I have an x s There what that means is I have a limit as X approaches positive Infinity of two over this is going to give you

one + 2 X2 you know I'll show all the work for you in case having trouble following me x 4 x 4 + 2x^2 over x 4 x^2 X2 are you guys satisfied with that one hopefully yeah okay good this gives you One so we're still at two that gives you one this this gives you 2x^2 and that gives you 1 we're now ready to take the limit ready let the X go to Infinity what happens to our two does it go to zero it didn't last time because we had an overx where's that go

where's that go where's that go what is it we've got a 2 over the < TK of 1 + 0 + 1 1 + 0 + 1 that's theun of 1 that's 1 1 + 1 is 2 we get 2 over 2 one did it change yeah so little bit yeah a little bit so that that little change there said we're not at zero anymore we have a horizontal ASM toote at now one so we're going at one for when we to Infinity how many you feel okay with what we've talked about so far Such

good stuff just kidding allergies are getting to me okay last one is it making sense so far can you follow that Shan you okay yeah oh that's nasty still sure you're okay with that one how about that one little tougher conjugate there's no Such thing as conjugate here there's nothing to distribute on the top and the bottom you to take it for just think about it for a no derivatives oh good try though when you anwers derivative good try this one of those 5% oh man yeah close no why it's going to you have a

negative under the square root ah so if I take Really big positive numbers subtract them from seven are they going to be positive or negative and they're under a square square root doesn't exist that was the easiest problem of the day come on no it doesn't exist so think about your limits I mean you can think of them like that you have a seven minus something huge 7us something huge is something really negative square Ro of Something negative is undefined therefore limit does not exist does that make sense now if I did this it's not

harder it's the same idea you have seven plus a really big number the seven plus a really big number square that is still a really big number it's Infinity all right so you you can think of that the only problems you really come up with is when you do things like infinity plus or minus infinity infinity Over Infinity uh that sort of thing gets sh you can't do those but if you know it's going to infinity or it's not going anywhere that's fine that's okay you can you can consider things like that there's no denominator

nothing to divide by you're good does that make sense sense soga infity would this no this is actually oh it's POS yes do you feel okay with our limits have I blown your minds yet Today so when you have issues where you're subtracting Infinities rationalize if you have a square root rationalize them if you don't think through the limit what it means to go to Infinity consider the type of problem that you have you already know pols any polinomial is going to go to positive or negative Infinity right we talked about that last time he

said well it follows really the leading coefficient or the leading term sorry the power leading Term so uh X Cub if it's it's going to go up or if it's negative X it's going to go down things like that happen so be prepared for those limits are there any questions before we continue all right