Stony Brook MAT 125 Spring 2015
Lecture 04: Limits and Continuity
February 11, 2015

Start   so last time we did limits as x approaches infinity and you want to make sure youre getting good at these so were gonna review just a little bit more so the standard idea, whats going on with infinity, theres two different things one is what makes you go to infinity, also known as vertical behavior what happens when you go out of infinity which is end behavior, another words
0:44say you have a graph that looks something like that those are 4s if you cant read my hand writing youre allowed to make fun of my handwriting
1:04so were looking for a limit as approaches -4 minus sign remember the minus side is this side of -4 so you get a curve you go towards where x is negative 4 and you keep going up up up continues going up it has an asymptote so f would equal positive infinity
1:35thats what we mean when we say x is minus 4 minus when you say x is -4 plus we know mean when your coming from this side of the -4 so numbers are just bigger then negative 4 but only a little bit so if you were thinking of plugging you would say im plugging in negative 4 plus a tiny amount or negative 4 minus a tiny amount if your on the other side so at negative 4 and going from this direction and you go down the curve
2:05so the limit as x approaches -4 ffrom the plus side, thats f of x equals minus infinity are those the same thing? no they are not because one is plus infinity one is negative infinity so the limit as x approaches -4
2:30of f of x does not exist and i know on some of the webassigns it wanted you to put none and instead wanted to put dne if you read the problem careflly it would tell you which one to put i know thats very frustrating dont know what to tell you alright well now whats going on if we go to positive 4
3:00well when we approach 4 from the minus side this side of 4, were going up to positive infinity so its positive infinity and what do we do if we do x approaches 4 from the other side well now were going down to minus infinity so were going down on the curve
3:37where you end up here you just keep going down and down deeper and deeper in the hole of darkest, scared because its sophmore year and then you get minus infinity so these two dont agree so once again
4:06so if we just ask for the limit as x approaches 4, we didnt give you a side it would be does not exist so thats, when you create a number you get out negative inifinty remember inifinty isnt a number, another words what you get out is the number keeps ggrowing
4:30it just keeps going up forever or down forever what happens if i want to know what happens after infinity well now if i do the limit x apporaches infinity im saying what haappens when im on the curve and i keep going out, bigger and bigger numbers of x
5:08and of course at minus infinity im also going to approach 0 notice their is no sign to an infinity limit like i said last week on monday there isnt a plus or minus side
5:30to infinity your either going to infinity or going to negative infinity because if theres a side you go to infinity and come back from another direction cant happen these two could agree cause youd get a curve that looks something like this now when you get plus 4 from the minus side you get plus infinity
6:03and when you approach four from the right side you get plus infinity so the limit as x approaches 4 would be infinity again so you ge the limit as x approaches 4 or the minus side of f of x youd get pi/infinity and youd get the limit as x approaches 4 from the plus side of x you would also get positive infinity therefore
6:32the limit as x approaches 4 is infiinty so far so good?
7:00now lets do some algebraic/numeric types so people were getting this a little confused at the end of class the other day so lets go through this again
7:38suppose you had that limit, limit as x goes to infinity 5x cubed minus 7x plus 1 on top 2x cubed plus 4x squared plus 1 on the bottom the numerator and the denominator
8:00so does everyone know what that is? we did the rule the other day its 5/2 now if you see this on the exam you can not just write 5/2 however you dont have to do a lot of work what you would need to do, first you would write as x approaches infinity the terms
8:48you write something as x approaches infinity the terms other than 5x cubed and 2x cubed are insignificant, so we can ignore them so this limit
9:01behaves like this if you write that youd get perfect credit, another words you dont have to divide by x cubed you could do all that stuff we did last time why write something like that because we dont know if you just wrote down 5/2 and youre not memorizing the rule and you have no clue whats going on or copy off the person neck to you and wrote 5/2 and wrote it larger
9:32when you are doing your exam you should remember you are giving your answers youre explaining yourself to the person on the other side going to be a ta whos going to grade 100's of these and is not going to be in the mood by the time he gets to ypurs for trying to figure out what the heck you are up to so you want to make sure its clear so that person could be me by the way, ill be grading one question so if im tired and im on paper 500
10:04i dont want you to try and get away with that so demonstrate your knowledge somehow you can demostrate your knowledge very simple but you essectially want to say, is when youre doing these kind of ratios the only thing that matters is the highest term on top, the highest power term on top in the numerator and the highest term in the denominator you can say top you can say bottom
10:32you can say treble and base whatever you want however we dont want to confuse this the limit as x is not going to infinity
11:25this is a totally different type of rational expression
11:34that works, lets see what happens, i just made this up i might make a mess of things but we will find out what do you do when you get something like this yea i like hearing thtat from the front so 25 plus 11 is 36 the square root of 36 is 6 is 0, so you plug is 5-5 and get 0 and then you cry
12:03you say what do i do? and rationalize how do i rationalize? i multiply the top and the bottom by the conjugate of that the conjugate of the square root x squared plus 11, minus 6 is the suqrae root of x squared plus 11 plus 6
12:30the idea of the conjugate is when you take the one binomial, a plus b and you multiply it by the conjugate a-b you get a suqared minus b suared so in this case, im gonna erase this, thats the idea of the conjugate im gonna erase and i erase alright so you got to get the conjgate so you ultiply this by this
13:03you get x squared plus 11 minus 36 and on the bottom dont try and combine terms just leave it alone doesnt matter
13:41so now x squared plus 11 minus 36 simplifies as x squared minus 25 youre still ignoring whats on the bottom notice if you still plug in 5 you still get 0 on top and 0 on the bottom
14:04so your life isnt perfect yet we can now factor the top
14:46alright now we cancel the x minus 5 and we get the limit as x approaches 5 whats left over, now you could now youre not gonna have a 0/0 problem, when you plug in 5 on top you get 10 doesnt matter you no longer have a 0/0 problem
15:01and whne you plug in 5 on the bottom 25 plus 11 is 36, 36 plus 6 is 12 okay you have homeowrks like webassign that looks like this i got that right, any mistakes?
16:06alright suppose we wanted to do this one so whats the limit as x approaches infinity of 1 plus e to the 1/x well as x approaches infinity 1/x approaches 0
16:34e to the 0 1 so as x approaches infinity this becomes 0, e to the 0 is 1 1+1 is 2 and what about when x approaches minus infinity its going to be the same thing okay
17:00easty ones again everybody see where thats ccoming from when x is infinity 1/infinty the limit of x approaches infinity is 1/x is 0 the limit of x approaches infintiy is 1/x
17:32thats a 0 thats because the bottom gets bigger and bigger so the whole thing get smaller as x approaches infinity 1/x appproaches 0 so that means this is becoming e to the 0 e to the 0 is because anything to the 0 is 1 so 1+1 is 2
18:00so far so good? what about when x approaches 0 well what is the limit as x approaches 0 from the plus side of 1/x well now the bottom is getting smaller and smaller so 1/x is getting larger and larger its staying the positive side of 0, so 1/x is positive so this is going to approach positive infinity
18:36and 1 plus infinity is infinity what about if you approach 0 from the minus side well now we get minus infinity why are we gonna get minus infinity
19:02well as we get closer and closer to 0, 1/x gets bigger and bigger but were using negative numbers so were gonna get closer to negative what is e to the negative infinity anybody know?
e to the -1 is e to the plus 1/x so as we approach positive infinity we are getting closer to 1 over e to the infinity so thats gonna approach 0 this is gonna approach 0 so the whole thing is gonna go to 1
19:34so if were doing this term 1/x as x goes towards 0 from the minus side this term will go close to negative infinty e to the negative infinty really means 1/ e to the positive infinity
20:01thats just 1 over a really big number so that just a really small number so this will get closer and closer to 0 the e term so the whole thing will approach 1 still see some glazed eyes out there
20:36everything we do is in radians all the trigonometry is in radians wen you cant get your trigonmetry stuff to work right first thing you do is check your calculator and see if its in degree mode put it in radian mode if your using some app on your phone figure out how to put it in radian mode cause if you email me ill tell you to check and see if you did it in degrees theres a webassign problem thats the tangent of something over the tangent of something
21:04your doing it in degrees you getting the wrong answer how do we feel about the whole limit to infinity thing?
21:34you guys dont get what hes asking, ill repeat this the best thing to understand whats going on with 1/x is look at the graph of 1/x the graph of 1/x looks like this this is the graph, y=1/x so what happens when x is a really big number?
when you have a really big number you put it in the denominator of a faction and its smaller
22:031/100. 1/1000, 1/a billion the odds of winning the powerball is 1/54 million thats basically 0, okay?
maybe not the power ball is tonight if i win you wont see me next class just so you know i will be figuring out something to do but ill be figuring it out from the bahamas
22:30and ill be thinkin gof all of you when x is very very large, this approaches 0 alrigt now what happens when x approaches from the plu side well now there putting very small numbers in the denominator so like wha is 1 over .00000001 thats 100 thousand so as the bottom gets closer and closer to 0 the whole thing gets pretty large
23:01so as you approach 0 from this side you get positive infinity similiarly what i like to say in the math world what happens when x approaches negativ infinity, its 0 because you get a huge denominator and the whole thing gets very small and you get 0, qhat happens when you approach 0 from the minus side you get 1/ very large number but its a very large negative number so its a small number so if you had 1/negative .00001
23:31you get negative 100 thousand so the limit as x appproaches 0 minus side of that will be minus infinity thats why were getting this you should store this in your head somewhere cause we like to do this kind of stuff keeps you on your toes, helps bring the class average down just kidding or that kind of stuff how we doing on the whole limits to infinity
24:01plus infinity minu infinity we sorta of understand this at some point okay time to learn something new, you got a question raise your hand because were leaving limits behind now not a lot of hand raising going on here that means you get it or your basically focusing on chemistry or something like that or candy crush
24:46so now we gonna discuss one of those important conceptual topics of calculus that drives people crazy, there certainly will be an exam question on this okay so we gonna talk about whats call continuity
25:02so we talk about, its very important in calculus to find functions a lot of whats going on requires a function to be continuous so what do when by continuous a continuous function is a function that doesnt have any breaks or holes in it, so if you were drawing it you never have to take the pencil off the paper how would we defind that mathematically, so a continuous function you know jut like that
25:31where a discontinuous function stops at some point and then picks up again you dont have to copy that but continuous means theres no holes in it, in a certain region there doesnt have to be no holes everywhere but if we say in some reason and its continuous in that region there no place where you take the pencil off the paper or the pen off the paper or the chalk of the chalk board so its smooth, it doesnt have hole
26:00it doesnt have breaks, it doesnt have asymptotes or any of those things, so how would we defined that well lets talk about the different types of breaks we can get we can get an asymptote, so we can draw something does somethin glike that its not continuous, right here whatever that number is a theres a break, so that function is discontinuous this is either called and infinite or essential discontinuity
26:35depends on where you went to school wither word is okay theres nothing you could do to fix that it is just not continuous it continuous everywhere else
27:15alright a second type of discontinuity looks something like that the function stops at some place and then picks up somewhere else
27:32and it jumps, so this is called a jump disconuity and again there is nothing you can do to fix that its just a hole its a jump and you cant,what i mean by fix youll see in a minute theres nothing you can do to fix the function to get rid of that problem so your continuous over here and your continuous over here
28:00but youre trying to get from here to here you have a discontinuity problem one thing that happens in calculus a lot of rules only apply if the function is continuous in some spots in some range, if its not continuous then we cant use the funtion, we cant use the rule third type of dicontinuity looks like that that is called a whole
28:31just a hole, a hole in a function or a rul of discontinuity so this one, is called a rule cause you can fix the problem kind like this is a pot hole in the road, you can fill a pothole however if it looks like this or looks like that, i suggest you take a different road
29:03this road is fixable, when it gets fixed depends on what the town is in the mood for what do these kind of things look like functionally everyone go the concept? these are the three main concepts of discontinuity either there just is not function, no value of the function theres a change in the value of the function or theres a missing value of the function
29:34and if you think about this whenyou think about something like this, what do we know about the limit, the limit does not exist or we look here and the limit doesnt exist at a because the left and the right side disagrree but here the limit does exist at a f of does not exist but the limit exist
30:06for example suppose this is the value like 2 this is a value like 5 then if you do the limit as x approaches a from the left side you get 2 the limit as x approaches a you get 5 since 2 does not equal 5 the limit does not exist here however the limit as x approaches a on the left side is 2 and the limit as x approaches the right side is 2
30:32therefore the limit exist you do not need f of x to exist f of a to exist, you only need both sides of the limit to exist so for continuity all we have to do is figure out how to get around that problem so we do a little erasing
31:08notice what goes on here the limit as x approaches a from the minus side of f of x is l l could be anything you want the limit as x approaches a from the plus side
31:36also equals L so far so good, so therefore the limit when x approaches a of f of x equals L and you say yes this function is not continuous cause theres a hole right there you can plug the hole, so what do i need?
32:00i need f of a to equal L so if both of these things aree true if the limit of f of x is L as x approaches a the limit is f of L and f of a equals L then the function is continuous so we will get a little more technical in a second but this is what we are looking for
32:30were looking for the left side limit is the same as the right side limit and that theres actually a value at a and that that value equals l becaue if the value was up here then it wouldnt be continuous so if you had a function
33:06you have a function like that the limit as x approaches a from the plus side and the minus side is 2 so the limit as x approaches a is 2 but f of a is 5 so its not continuous so you would need both the function and the limit to have the same value so theres a slightly more technical verzison of this which
33:32ill right down later, its in the book but this is basically what we are looking for, theres 3 things we have to figure out so lets do an example
34:04suppose we have something like this, this is a piecewise function because its defined in pieces so the question is, is this continuous can you try and do this without graphing
34:35what is the limit as x approaches 2 from the minus side the limit as x approaches 2 from the minus side will be values less than 2 1.99999 when i plug in a value less then 2 i get 4-1 less than 3 what happens when i plug in a value just greater than 2
35:04when i plug in something just greater then 2, i get omething at this part of the curve 6+5 is 11 these two do not agree that means that that function is jumping its doing something like that
35:31so the limit from the left doesnt equal the limit from the right, theres no limit there so this function is not continuous at x equals 2?
just so you may know you might need to write this at some point on your exam polynomials are continuous everywhere so you can look at this and say, this is a polynomial so its continuous everywhere less than or equal to 2 this is a polynomial so its continuous everywhere greater than 2
36:02so your only problem is at 2 and at 2 the limit does not exist so therefore the function is not continuous alright lets take a slight variatiion on this
36:47now we have almost the same function we have 3x+5 when x is greater then 2 and 2x+7 when x is less than 2 so is this function continuous
37:02okay lets figure it out, well the limit when x approaches 2 from the minus side of f of x is 4 plus 7 is 11 and the limit when x approaches 2 from the minus side fo x 6 plus 5 is 11 so notice what youre really doing youre plugging 2 into both of the equations
37:30and see if you get the same number and look you get 11 both times so the limit wen x approaches 2 of f of x is 11 so is it continuous at 2?
no why not?
well whats f of 2 notice i wrote greater than 2 and less than 2 did not write equal to 2, i did that on purpose
38:03remmeber your gonna take the exam at 8:45 at night so you want to make sure you catch those things so there is no f of 2 so once again that function is not continuous at f equals 2 continuous everywhere else but not at 2 okay slightly more annoying
38:30that satisfies the first criteria when the limit existed if there wasnt a second criteria, were not quit there lets do a more annoying one
39:12so what is the limit as x approaches 2 from the minus side when i plug in 2 i get 4 plus 7 is 11 and whats the limit ofx approaches 2 from the positive side its equal to 11
39:32so therefore the limit when x approaches 2 f of x is 11 but f of 2 is 10 so we satisfied almost everything we need, the limit exist f of 2 exist but they dont equal eachother this is an example of the type where you have a hole
40:00and then a little dot in another spot it looks something like that when you get to 2 the limit is 11 but the function is only equal to 10 at exactly 2 when x is less than 2 you come from the left sde when x is
40:33greater than 2 you come from the right side cause less than 2, greater than 2 now lets do one that works a new function so you dont get bored
41:13suppose we put this on the test and we said is f of x continuous at x equals 2 well first thing you say is f of x is continuous everywhere other than x equals 2 because they are polynomials polynomials are continuous everywhere your only spot
41:33cause its right where the equal shifts alright i we plug 2 in the top of the equation, well lets get the limits heres how you check, first in your head you take 2 you plug it in and you get 4 plus 3 is 7 2 and plus it in you get 12 minus 5 is 7 you get 7, 7 they agree, theres no missing equal sign so these things are going to be continuous, so how are we gonna show this
42:00so professor and ta give you full credit you say the limit as x approaches a from the minus side of f of x thats this branch is 12 minus 5 is 7 and the limit when x approaches 2 from the plus side of this branch is also 7 therefore
42:32the limit of f of x is 2 equals 7 and you say f of 2 equals 7 therefore its continuous at x equals 2 that would be a good enough answeer
43:00and by the way if you want to you can abreviate continuous cts so you can save yourself from writing out a long word and replacing it with a short word math is filled with that, thats why we dont write therefore we put the 3 dots does anyone know where that came from? if i made something up youd believe it okay so far so good?
because this is where is
43:31where x equals 2 when x equals 2 i just plug in 2 and i get 4 plus 3 is 7 rememebr on this one, i dont have a problem where x equals 2 but here i do and there i do so watch for that, less than greater than or equal to sign thats one of the ways we get you guys
44:00so lets see, whats a typical type of question alright well give you guys one to work oj
44:54find the value c so f of x is continuos on the interval minus infinity
45:00plus infinty, you want to know why you write those with perenthasis and not square brackets because you cant actually have infinity, you cant actually have a square braket dont use a square bracket, use a parenthesis find the value c where f of x is continuos everywhere c is cx squard plus 4x when x is greater than or equal to 3 and 5x plus c when x is less than 3
48:20alright thats long enough so in order for this to be continuous the limit from the left side has to equal the limit from the right side
48:30and they both have to equal f of 3 so the limit from the left side well we say when x is less than 3 5 times 3 is 15 plus c and the limit from the right side is you plug in 3
49:00and you get 9 c plus 12 thats also what f of 3 equals by the way so in order for these to be continuous this has to equal that so 15 plus c has to equal 9c plus 12 so you do a little algebra and you get is 3/8
49:34well i may made a mistake you never know you take 3 and you plug it in here c times 3 square is 9, so 9c 4 times 3 is 12 alright lets give you one thats more annoying
50:56alright we have 3 parts of this function
51:00we have x squared minus 4 over x minus 2 is less than 2 ax squared minus bx plus 3 when x is between 2 and 3 and 2x-a+b when x is greater than or equal to 3 find a and b to make this continuous everywherre good exam question in order for this to be continuous the limit as x approaches 2 from the minus side has to equal x minus 2 from the plus side
51:33so the limit x approaches 2 from the minus side of f squared minus 4 over x-2 problem is you plug in 2 you get 0/0 to think about this and say wait i can factor the top
52:00since its not actually 2 we can cancel this and get 4 so the limit as x approaches 2 from the minus side of this equation is 4 and what about the plus side that equals a times 2 squared so 4a minus b times 2
52:31so this limit has to equal this limit, has to equal f so 4a minus 2b plus 3 has to equal 4 4a minus 2b has to equal 1 so far so good now lets do the same as x approahes 3 limit of x approaches 3 from the minus side
53:02is the middle function now were plugging in 3 not 2 and the limit as x approaches 3 from the plus side of f of x equals 6 minus a plus b so when we plug those together we get 9a
53:373b plus b has to equal 6 minus a plus b so 10a minus 4b has to equal 3 so there you go we got 2 equations and 2 variables so we got 2a minus 2b equals 1
54:00and 10a minus 4b equals 3 a is -5/2 lets see what id do, id multiply this equation by negative 2 and i would get negative 8a plus 4b is negative 2 made a mistake there and 10a minus 4b equals 3
54:31and you get 2a equals 1, a equals 1 a excuse me now that i know what a is i can go bac and find what b is see what i did, should be able to solve 2 eqautions and 2 variables now that i know its a half i know 10 times 1/ minus 4b equals 3 so minus 4b equals minus 2
55:01b also equals a half so i add these 2 equations together i get minus 8a and plus 4b plus 4 b and minus 4b cancels well you can do substitution but substitution is messy you can solve this however you want what did i do here, i took this equation
55:31i multiply it by negative 2 so that becomes 8a minus 2 times minus 2 is 4b minus 2 times 1 is -1 and i took the other equation and put it underneath now i add them together 10 minus 8 is positive 2 the b's cancel and then you get 1 2a is 1 thats the methond of symetanious equations like i said did memories from 9th grade that was a long time ago for some of you guys
56:00longer though for me 9th grade was a while ago i can only count that high its an limit approaching infinity but thats some math word do you want to know why i used negative 2 theres a bunch you couldve done but the easiest you could do,a another you could do you could divide 2 by something and you get b
56:31and you get b alone, you have a lot f techniques to decide at that point everyone understand what i did sould i do one more like this?
who votes for 1 more who does not vote, whos had enough thats a tough call, youve had enough lets do one more
57:04ill make it slightly easy
57:50okay why dont we do this as a team so we have this really great function
58:02ax+b-1 when x is less than 1 x squared minus 3ax-2b when x is greater than 1 but less than 2 and 4ax plus b plus 3 when x is greater than or equal to 2 do you know where these equations show up? they show up in like theyll show up when you have a change going on for example a business function youll have cost or expenses and
58:30this is if youre running 1 machine this is if youre running 2 machines this is if your running more than 2 machines things like that so you see this a lot in operations i cant speak for biology cause i took biology in 9th grade and 9th grade was not recently i took biology from about a mile from here i took calculus here
59:03alright so whats the limit when x approaches 1 from the minus side well plug n 1 in the top branch and you get a plus b minus 1 the limit when x approaches 1 from the plus side go to the middle branch and you get 1+3a-2b
59:31these have to equal eachother because as you approach 1 from both sides you have to get to the same spot a+b-1 has to equal 1+3a-2b do a little magic algebra and you get 2a-3b equals -2 alright now lets repeat for the bottom 2 things
60:03the limit as x approaches 2 from the minus side you plug in 2 you get 4 plus 6a-2b and when you appraoch 2 from the plus side
60:32you get a a plus 2 plus 3 and these have to equal eachother so 4 plus 6a minus 2b has to equal 8a plus b plus 3 so 2a plus 3b has to equal 1, wow that came out well didnt it
61:05really proud on myself, so 2a minus 3b minus 2\ and 2a plus 3b equals 1 i hope you can solve this you simply add these two equations together you get 4a equals negative 1 a is negative a qauter
61:42howd we do?
so now you got a as a qaurter so now you go back and find b so 2 times a quarter plus 3b equals 1, 3b is a haldf
62:02b is 1/6