Stony Brook MAT 125 Spring 2015
Lecture 03: Limits, Limits involving infinity
February 9, 2015

Start   So last time we learned about limits and played around with them a bit okay lets makes sure you can do some of these then were gonna do some other stuff, so the limit laws
0:31remember that means we can add them subtract them and multiply and all that stuff you can have some very entertaining problems with limita so heres one
1:14okay that says the liimit as x approaches 25 of 5 minus the square root of x over 25x minus x squared now if you plug in 25 you get 5-5 on top thats 0
1:30you get 25 minus 25 minus 25 squared on the bottom thats also 0 so thats gonna cause you a problem so what are you going to do?
im hearing some good answers, can do some rationalizing you can do some factoring you can do rationalizing and factoring ill justpick one, so what do i mean by rationalize?
notice the top is 5- the square root of x, the square root running around
2:03you almost cerntain have to multiply the top and bottom by the conjugate the conjugate of a minus b is a plus b and the conjugate of a plus b is a minus b the conjugate of 5-x is 5 plus the square root of x
2:31so why did we need the conjugate remember conjugate of a-b is a plus b and the other way around so when you multiply together the conjugates what happens is you get a squared minus b squared what you all know in other words the middle term goes away you squared rooted the problem so you multiply 5- the square root of x times 5 plus the square root of x we have the terms without the radical anymore
3:02on the top were gonna have a problem on the bottom but were not gonna have a problem on the top so lets see what that looks like 5 times 5 is 25 minus sqaure root of x times square root of x is x that coming through okay?
3:30started talking into the cord at some times lets see how that comes out on the video now that doesnt help because when i still plug in i get 0 on the top and i still get 0 on the bottom but now i can factor it which you couldve done before but it doesnt really matter so leave the top alone notice i didnt do something like multiplying this out thats not fun
4:07when you have these problems with conjugates dont multiply out the bottom or the top youre just gonna do some canceling thats what you do with conjugate notice the 25-x cancel thats the term where we had the problem thats where we get the 0's from so now
4:31you have the limits x approaches 25 of 1/x times 5 plus the square root of x notice its not 0 on top its a 1 now w plug in 25
5:03i get 25-10 which is 250, which is 1/250 when we do the limit notice i keep saying im plugging in 25 but what were really doing is youre not plugging in 25 remember youre plugging in a number really close to 25 24.9999 but as far as it concerns we want to see what happens if it were 25 so thats how were able toc cancel this
5:31we cant actually cancel that x 25 because you cancel to 0/0 were not actually plugging in 25 these two numbers are very tiny, but becasue they are not 0 you can cancel them itd be the same so you get 1 so now with the kind of limit stuff we did last time just as a heads up the midterm is in 2 weeks 2 weeks from thursday so so thats very soon, you guys may say wow thats very soon and so are wee
6:02there cant be a lot on the midterm because we havemt done much yet certainly are for the clas so the midterm is gonna have limits on it and its gonna have some other stuff im doing and thats about it so make sure you can handle these limits im working on setting up a review class but i dont know when yet it will be before the midterm i would have it after the midterm
6:31and i told you guys this on the first day, we have the midterm 2 weeks of class, spring break a week of class and then another midterm again we dont have a lot on the second midterm but that means well have to learn a lot for the final
7:01so lets look at some other limits stuff, so if you were wondering where we are have any of you bought the book or following through the book were now up to 2.5 oh you also have make up classs
7:33just thought i should let you know it will be at 4 oclock well discuss that when the time comes im working to make that an object of class but its not really up to me we all know what option means right?
option means people show up if im lucky, unfortunately right now we have class so dont get excited now
8:08so now were doing limits involving infinity this is the fun stuff what do we mean by infinity get x as large as it can it gets bigger and bigger numbers and what happens to the function
8:39thats a function that looks like that -x approaches 5 and this is your function notice what happens, 5 is the vertical asypmtote when it gets really close to 5, the function goes straight up so you write this as limit
9:01when x approache 5, minus 5 f of x, we would say thats 5 to infinity so the left side limit look at the graph and say yea thats infinity, and the right side is also infinity therefore the limit of f of x is 5
9:31is in infinity you dont neccesarily have to write the function but you dont want to get confused okay if you write the minus symbol you dont do anything you assume you get infinity you assume positive infinity you dont have to write the plus sign i advise you put them in and the minus signs cause we would know you know what youre talking about
10:00notice its not that it doesnt exist its that you get infinity but then you can say there isnt actually a value its infiniyt its a little tricky what we mean when a limit does not exist what a limit does not exist means you go from one side and you go from the other side and you dont get the same answer so the proper answer for this would be infinity now what if on the other hand
10:34i have that now when x approaches 5 x apporaches 5 from the minus side you get plus inifinity when x approaches 5 from the plus side
11:01were going down so we get minus infinity okay so what happens at 5? the limit does not exist what a verticle asymptote tells you that it has an infinite behavior its getting bigger and bigger plug in numbers closer to 5 you get here
11:31you get closer and bigger and bigger answers and thats gonna get you 4.99999 and then the otheer side you earn a negative answer you get negative infinity number notice they dont match now lets go back and look at this for a second what happens when a limit goes x to infinity
12:06now when x goes towards infinity youre going this way and we look at the horizontal asymptote at the x axis so you say this limit is 0 that make sense?
okay and x gets bigger and bigger
12:30trillion and trillion and so on were getting closer and closer to 0 and the same on the other side so here the limit as x approaches negative infinity is also 0 and notice you cant have two sides cant be on the other side of infinity and here again as x approaches infinity positive infinity you get to 0
13:03as x approaches negative infinity you get to 0 so that if you have a little less then 0 or more then 0 its still 0 theres a few things about infinity one is you see what happens when you go to infinity the other you get to a number and approach infinity that make sense? one is infinity on the x side the other is infinity on the y side
13:31alright so lets lets look at some important stuff to understand say we have f of x equals 1/x so lets start plugging in numbers for x
14:03if we have x is 100 you get 1/100 when x is 10000 you get 1/1000 when you get x equals a million you get 1/a million so what happens when x gets to big it approaches 0, okay
14:41what about when x is negative a million negative a billion, youre still gonna get 0 you just get 1/negative billion which is basically 0 so it actually doesnt matter if this is plus infinity or netive infinty so you say when x approached positive infinity or negative infinity
15:02f of x is 0 and thats because as you plug in bigger and bigger numbers in the denominator it gets very very small so if we were to graph that thats approximately what 1/x looks like so notice as x gets very large it gets closer and closer to 0 it doesnt matter what irection youre going
15:42what about when x gets very small when x goes to 1/100 or 1/1000 well okay f of x is 1/x f of 1/10
16:02is 1/10 which is 10 f of 1000 is 1/1/1000 which is 1000 and so on, when x is 1/10000000 1/1/million is a million
16:38so on and you can see it on the graph as x gets closer to 0 iyts going to be very very big its going to blow up towards infinity so the limit as x approaches0
17:04of f of x is infinity, remember this one ok this is the stuff that tricks you guys up so if x is very veru close to 0 and a positive numbr you get positive infinity and what about if i get close to 0 and its a negative number well if you a negative a million you would have 1/ negative a million would be negative a million which would be a very negative number so if x approaches the other side
17:39you get negative 10000000 well maybe another class we would have a microphone you guys can all hear me back there okay?
18:05if you cant hear me nod your head nobody nodded their heads, there we go so some of this stuff goes on in infinity so remmeber its very important to remember when you get to 0 0 in the denominator the whoe thing becomes very large you put 0 in for x, you get a function that is very big
18:51suppose you have something like thta
19:25thats says as the limit approaches 3 from the plus side
19:312/x+# so first you say to yoursef so what does that ,ean 3 from the plus side, that means a number just a little bigger then 3 so if you plug in a number just a little bigger then 3 3.000001 and you subtract 3 youll get a very small number but its gonna be positive that means this whole thing would be positive and the bottom is getting closer and closer to 0, the bottom is getting pretty large so that would equal positive infinity
20:06now what about if you do from the other side well now youre putting in a number just less than 3 so if you take a number a little less then 3, you take the number and subtract from 0 but youre gonna get a negative number youll get 2.99999
20:30you subtract 3 you get negative .0001 if you do 2.99999 you subtract 3 you get .00001 when you put it in the denominator the whole thing becomes very large, when its negative so you get negative infinity so this is what i ask you, what is the limit of x approaches 3
21:04does not exist isnt that fun this is the whole does not exist thing, you have been wondering you whole life what this means or maybe not but now you know alright so some other stuff involving limits and infinity
21:31heres something we did in precalc, and we do it again suppose you wanted to do the limit as x approaches infinity
22:03you get 3x-1/2x+5 so this you have horizontal asymptotes this is gonna come out 3/2 so how do we know?
well when you plug in infinity you get infinity on the top and infinity on the bottom, thats not very helpful
22:33what could you do?
got any ideas? go ahead so thats our short hand, how we get our answer question how do we know its right dee how the highest power on the top is x and the highest power in the bottom is x divide all the terms by x
23:03so you would get 3x/x minus 1/x 2x/x plus 5/x x is a number so we cancel these
23:39s o you get 3-1/x and 2+5/x now as x goes to infinity, what happens with te 1/x approaches 0 and 5/x also approaches 0
24:02so you just get 3/2 we taught you a rule and this is where the rule comes from got it lets do another one
24:47again that says the limit as x approaches infinity on top you have 7x squared minus 8x plus 2 on the bottom you have 4x squared plus 5x plus 3
25:01and a lot of you already know how this is gonna come out cause you did precalc or you did it before you had precalc someone of you did this when you were little children, kidegarden, maybe not the highest term of the top is x squared the highest term on the bottom is x squared so multiply it by x squared
25:30you get 7x squared over x squared minus 8x over x squared plus 2 over x squared and on the botom you get 4x squared over x squared plus 5x over x squared plus 3/x squared, you guys can see where this is headed
26:00and we get some canceling on top you get 7 minus 8/x plus 2 over x squared, on the bottom 4 plus 5/x plus 3/x squared what happens if x goes to infinity
26:300 0 0 0, the whole thing just equals 7/4 so again all that happens is all those terms all these terms cancel if they dont have x squared in them these terms dont matter the main terms well we got 7x sqaured and 4 x squared
27:02the x squared cancel and i get 7/4 why do the other terms matter well if you took out your calculatorsz youn plug in something like 1000 this would be 7 million, 7000 squared is a 7 million thats only 8000, 2000, 5000, thats a 3 1000 dollars is a small factor of 1 million, if i gave you 1000 and asked for a million back you still pretty much have a million dollars youd have 999 thousand dollars but you know
27:35you put in a million and you sqaure it you get a trillion you get a trllion dollars you take away a million dollars you still have a trillion dollars so at some point the big number becomes some much bigger then the other numbers that the big numbers dont matter the only thing that matters is the highest power got that? cause all the other powers
28:03so if you win the power ball on wednesday is 450 million dollars and then i take away a thousand dollars youre still gonna have 450 dollars, and were gonna be really good friends now of course you have the problem where the powers are not the same
28:31there becomes a point where it actually pays to buy a ticket in fact i did the math and its 80 million bucks a lo bigger then that what if we had
29:09okay what if we had something like this so the limit of x to infinity is on top 6x sqauered minus 3x plus 5 on the bottom you have 4x cubed plus 8x squared plus 11x plus 1 can you guys read my hanwriting all the way back there?
29:40alright so now lets do our trick notice the highest power on top is x squared and the highest power onthe bottom is x cubed so do you think you should divide everything by x squared or x cubed x squared, waiting to see what i do yea thats what i thought, lets do x cubed
30:07the reason you want the highest of the 2 powers because when you divide by the powers you get 0 thats what youre trying to do youre trying to get rid of things so divide everyting by x cubed so you get 6 xsquared/ x cubed minus 3x/ x cubed plus 5 over x cubed
30:42okay everything is over x cubed so now we have lots of canceling
31:02and we get the limits as x goes to infinity of 6/x minus 3/x suqared plus 5/x cubed on the bottom you get 4 plus 8/x plus 11/x squared plus 4/ x cubed
31:31okay so now lets get the limit as x goes to infinity and all the top terms become 0 and all these terms are going to be 0 so this will then equal to 0/4 and whats 0 over 4 equal to?
so if you have a higher power on the bottom then the top
32:00then you divide by the x term youre gonna cancel on the terms on the top and just get 0's you get something on the bottom which is 0/something which is 0 remember this when you have a fraction and you have 0 on top its 0 when you have 0 on the bottom its undefined, later you would be solving derivatives and setting it equal to 0 and sometimes all you have to do is set the numerator equal to 0
32:33because a fraction is 0 when the numerator is 0 anticipating you raising your hand a few weeks from now saying why are you only setting the numerator equal to 0 the answer is the fraction is 0 when the numerator is 0 so otice what happens you have a higher power on the bottom so infinity this whole thing became 0 what happens when you get the higher power on top well lets see
33:24okay lets take a second and lets see what happens to this one
33:30so the limit of x goes to infinity on top we have 3x cubed minus 8x squared plus 5 on the bottom we have 2x squared plus 4x plus 1 so we have a higher power on top then the bottom alright thats long enough., now lets divided everything by x cubed were gonna get 3x cube dover x cubed
34:01minus 8x squared over x cubed plus 5 / x cubed all over lots of canceling you get 3 minus 8/x plus 5/x cubed
34:34on the bottom you get 2/x 4/x squared 1/x cubed now whats gonna happen whe you get 3 on top you get 0 on the bottom but what does that mean? its not undefined the question is can we figure out if its gonna be positive infinity or negative infinity
35:01the answer is yes this is a 3 so its a positive number these are gonna all be 0 so thats gonna look ike positive infiniyt s=does that show up on camera guys this is very exciting that im videoing all this next year at the oscars ill get something and wear it at the red carpet
35:49so this is gonna give us a rule so heres the good stuff okay?
36:00when you have a ration function when you have polynomial on top like weve been having and a polynomial on the bottom the highest degree the highest term on the top a x to the m whatever m is and the highest term on the bottom is b x to the n, whatever n is
36:30it doesnt matter what the other terms are it only matters what the highest term is but if you see you can cancel the other terms so if the highest of the term on top has a higher degree than the term on the bottom and with the canceling were gonna end up with a number on top and a 0 on the bottom this is going to go to infiniyt okay thats gonna be the limit sometimes its negative infinity so you have to be careful how would you know? youd have to look at the sign at whats left
37:05so another words if a is a positive number youll go to positive infinity if its a negative it will go to negative infinity remember all the other terms turn into 0, play around with this a bit in your calculator and yulll se what if you have the highest term on the bottom it would be 0, we did one of those because you have a bigger term on the bottom
37:31youre gonna nd up with a number on the bottom and 0 on top so this is going to be 0 and if the powers are the same then you get a/b you did this before this is how you find horizontal asymptote what is a horizontal asymptote \ a horizontal asymptote is accentially what happens as x approaches infinity or negative infinity
38:05its the end behavior its all figured out its the limts
39:00so we have 3 different rational expressions ill give you guys a second to copy those down the first one we have 6 x cubed minus 8x plus 1 over 5 x squared minus 3x plus 2 so we have a higher power on top the second one we have 6x squared minus 8x plus 1 and the bottom we have 5x cubed minus 3x plus 2
39:32so we have a higher power on the bottom and the 3rd is 6x cubed plus 8x plus 1 and 5 x cuved minus 3x squared plus 2 the powers are the same so as x approaches infinity this first 1 would equal infinity we have a higher power on top thats what our rule tells us
40:02how do we know its plus infinity becaauseoff 6, if it was minus 6x cubed it would be minus whats gonna happen is when x is really big like a million this term would dominate all the other terms it would so muxh bigger the other terms wouldnt even matter so the bigger power on the bottom the higher degree on the bottom so this will go to 0
40:33the last one since we have the same degree on the top and the bottom its gonna aprroach 6/5 more stuff with infinity, were almost done with the infinity stuff
41:10if you want to know when to use this or show up, on a biology test or something like tha some kind of reaction, wondering the wrong term of the reaction so you have some chemical and its bubbling away and you get
41:31but at some point nothing is happening amnymore and you want to figure out what happens to this reaction over a long period of time so infinity might as well be infinity at some point things arent happening anymore so you can approach a number or if i keep going id firgure out the number be 0 so bionical growth at some point some things just things just keep getting bigger and bigger and then then they explode
42:00sometimes they become a flat number because you just get more and more people on earth some point point wed have an infinity number on earth and then it gets crowded we may have a problem before then but maybe not okay we would have an infinity number of people on earth probably not thats where this shows up so youll see this in chemistry
42:34maybe in biology so other stuff to know about infinitiesz remember when i told you the term e to the x, we use that alot in calculus i know how much you love that so e to the x looks like this if the function f of x equals e to the x
43:04so 3 important things to notice first of all e to the x is always positive never negative, its 0, always positive take a possitive number and no matter whapower you raise it to positive or negative or a fraction or 0 whatever always gonna get a positive number
43:31second the limits as x goes to infinity is infinity it just keeps getting bigger and bigger this is the whole exponetial growth thing it just keeps on growing
44:00and what about when x approaches negative infinity that way well you get e to negative infiniyt you get a negative very big number and thats gonna approach 0 remember what a negative power means 1 over, so e to the negative 1000 that is e to the 1/ positive 1000 so 1 over a very big number is 0
44:36okay lets figure out a couple limits
45:28okay take a second and see if you can figure that out
45:48you all see that, limit as x approaches infinity x+2 on the top square root of 9x squared plus 1 on the bottom
46:36we have to square root it, isnt that annoying, why do we have to do that think about what happens when x gets really big when x gets really bigger what affect does it have on x plus 2 plus 2 doesnt do much of anything does it?
a million plus 2, thats still a big number on the bottom what do you think the 1 does
47:019x squared plus 1 plug in a really big number the 1 becomes irelevant so this is gonna act just like x on top and the square root of 9x squared on the bottom okay because the plus 1 and the plus 2 dont really do anything
47:31now the square root of x squared is just 3 x technically its absolute value of x but whatever you get 1/3
48:14now we have to learn so by and example
48:36you knew sooner or later we needed to get trigonometry involved you could go the rest of your life without trigonometry but nope here it is the limit of x to infinity of sinx/x squared
49:41suppose you wanted to find the limit as x approaches 0 square root of x sin 1/x
50:43well ill give you a clue okay you have sin 1/x sin /x is kind of weird because because 1/0 is infinity so again its gonna be isolating btu what do we know about sin its always 1 and -1
51:06so we can say, sorry let me rewrite this sin of x is always been 1 and -1 thats going to be able to the limit of x approaches 0 the square root of x times 1
51:31and its always going to be greater than the limit as x approaches 0, of square root of x times -1 the limit as x approaches 0 of squared root of x times 1, is just 0 because whats the square root of 0 0 0x0 makes 0
52:00so the limit as x approaches 0 square root of x sin over x is always gonna be 0 the limit as x approaches 0 of square x sin 1/x is always going to be greater then 0
52:30so therefore the limit as x approaches 0 is the square root of x sin 1/x is 0 lots of infinite stuff
53:15either im really good or you guys are just really tired lets just do a couple more not to much
53:55i want you guys to try these two on your own for a minute see if you can solve them
54:16alright lets do it do confuse this as x goes to infinity that is not a limit as x goes to infinity thats the limit of x goes to -1 you cant use that power rule thats not whats going on here, however
54:32plugging negative 1 on top is 0 plug in -1 on the bottom you get 0 so lets see if we can factor something why yes we can on the top we get x plus 1 times x plus 1
55:02on the bottom we get x squared minus 1 times x squared pllus 1 but wait we can factor this some more x squared minus 1 is x minus 1 times x plus 1 and now we an cancel those x+1 terms
55:30and we would get the limit azs x approaches negative 1 x+1 on top and x-1 times x squared plus 1 on the bottom and now you plug in negative 1 what happens?
you get 0 on top you get not 0 on the bottom so this equals 0
56:02okay questions?
that one was painful how did we do on that one?
alright what about the other one well we plug in -4 we get 0 on the top and 0 on the bottom any ideas why dont we combine the two terms on top and find a common denominator
56:36so on the left side we get x over 4x and on the left side we get 4/4x you get 4+x
57:00over 4x all over 4+x okay the 4+x now cancel and you get the limit as x approaches 4 of 1 over 4x now when we plug in4 we get
57:32-1/16 okay thats all, you cant go from there