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AP Calc 1.2a

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Part 1 of 2 for chapter 1.2

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hello everybody we are in our second calculus truly calculus video lyrics we are going to talk today about finding limits graphically and numerically and our objectives for his lesson going to be to estimate a limit using a numerical or graphical approach and we also want to understand when limit do and you stop exist so let's get started  the first thing the first question like to address is what is  a limit  what is that mean what is element let's start to answer this by considering an example  so we're going to consider the song shine f of x equals X cubed part of me still one -1 / X - 1 and that's a cubed right there in case you're having a hard time seeing that okay and what we're going to do next is going to simplify this if you guys remember from precalculus the top of this traction can actually be simplified if we Factor it if you look at your Apollo meal cheap cheap that I gave you in class and you'll know that X cubed minus one can actually be expanded out and to the factors x minus one outside x squared plus X Plus 1 and then on the bottom we still have access to -1 and now we know if x minus 1 over x minus 1 week and across this off and we are left with this new trinomial  okay that's all great and good but remember the way that domains work is the domain of this trinomial still  cannot have x equal to one because if we look at our original function we had x -1 on the bottom if we had one in here for x and get 0 on the bottom and that's not allowed to even though we changed some things and we simplify are domain still has to obey original function okay that's all stuff we know from precalculus here's to calculus  we want to  understand what this graph is going to look like now that we understand X Canticle one even though we have this problem so the first approach were going to take to do this is called numerical and the numerical approach life is really just using the table function in your calculator so let's do that what we're going to do here is kind of right that's all the better  move it over maybe a little bit this way yeah okay oh you know what let's move it okay first thing we're going to do is good where y equals is really hard for you guys to see  there's not but that's still pretty bright let's try Vanessa turn me technical difficulties  let's see if that's little bit easier no  let's try I'll hold it at in angle like that for now yes that's better okay here we go we got here so we're going to type our original function X cubed minus 1 over x minus 1 so let's do that Elsa Atlanta low price Whitey quotes and then press the first one and we get that fraction bar will put X ^ 3 - 1 all over x minus one  okay and we're going to Grass this and when we do that we see that it makes that nice problem makes sense because we understood that this trinomial is the same simplified expression now what we're going to do is  we're going to look at our values as they get close what happened because you don't really know how to deal with this yet we know that our domain can't be equal to a we don't understand what happens to a graph or throw value given an equal one so let's find out let's go to our table set let's make sure table set I set up for this problem so we're going to press the second and the Morgan press window  and we're going to make sure that our table start how about at negative 0 play five or take a look at that and then let's make our table change 0.01 and that will help us get really close to your you know what  let's make it .1 how about that that'll be know that better let me go change and now we'll go to our table and we'll press the second graph to do that  and here we are okay  so we can see that as our values get down point 5.44 3.2.1 are we get pretty close to be looking at one right card meet let's make sure we do that  okay so let's go back to tables that actually in let's change my so we're going to go a sec window again  and we're going to start our table let's say at causative  let's say points  seven five and now I'm will make our change maybe point  05 so be a little bit closer for going to press the second table again okay I just going to help us visualize just a little bit better c a z plug-in Oliver values got there we go it's a little bit easier for you to see okay are we started here at .75 we get close close close close close to one and we approach it looks like about 3 send our table one more time just to make sure that we're doing the right thing here and at where you really are right that it is approaching 3 so will press second window again the time of try starting at point nine how about that and then we'll change by point one will be a lot closer now again okay this is my snake so as we see  three scroll all the way down I'm trying to put one in the middle of her Fable there we see that as our values  increase to one that come in from the from the right direction and they  approach one we see they get very very close to 3 we of 2.9 and I'm very a decrease towards one as they come in from the left we can see that they are getting also very very very close to drink okay interesting  so our numerical approach we could just copy down some of those values a really don't have to write anything else for that you can just write a note in there for yourself under numerical let's move that stack a little bit can I know that you can use the table function in your calculator any looked at your value the increase in Direction and a decrease in Direction and see what it is right iron for the graphical well again if we take a look at our grass oh my gosh I totally calm down look at that better  from the graphical approach to be pressed graph let's try if it just too bright graphical approach with you have that trinomial paragraph is actually going to have the following little hiccup in it set up a sketch here and we know we're crossing at 1  and we know we get to about three like I was we discovered an hour chimerical approach  we actually have an error don't lean so we do is leave Sketcher crap like we would guess that Progressive but I we approach when we get really close to 3 but then we have this open circle at 3 open it doesn't look very good looks initiated in but I promise it's open we have an open circle at 3 because again X cannot equal ones we actually don't have that value their but we know that the limit as we go nrx values this direction and we go in RX I use this direction it gets closer and closer and closer to 3 as we saw what are numerical approached and Illustrated graphically now  so what we can do with this information is  we can say even though ex can't be one as we get very very close to 1 get very very very close to 3  enso in Calculus BC  that is the same thing as the limit add axe approaches 1  for the function cubed minus 1 over x minus 1 is equal to 3 the limit so the X values are part of me the razor function of protein as X approaches that's what that is saying  okay that's where the term limit comes from we just we can't even when we can't have certain values and crafts what we can do is quote on quote take the limits and fine what those numbers get close to alright that answer the question what is the limit next thing we're going to talk about it setting a limit estimating a limit an example of that  is going to begin with us evaluating the function f of x is equal to X over rid of x + 1 - 1  axe several X values near zero estimate  are the limit as X approaches 0 for the function okay  so what we're going to do next is going to be very very where's who what we just did we have our two approaches that we're focusing on today numerical and graphical so did you first remember that's how we just use the table function in R calculator and we're going to look at our values and our table as they get closer for certain zero cell have so  there we go that's not there bad let's type that in we have  again do you like the function by alpha y equals  Matt fraction bar if I said function burn it was not what I meant to say my bad Swiss ax over and then we can put radical X where is our radical there is Free Press second  and then  and looking to type in x last one and not and then we do minus one and if we graft us  let's first make turn our table is set up to help us see this clearly and again we want to get not close to one this time going to start 9/6 do you want to see closer to zero so we're going to start at let's say look up .1 or how about .05 and then we're going to go by .0 and now we're going to press the second graph to look at our table  okay we go all the way down on looks like maybe we got a little started a little too high there so let's go back let's press second window and let's not start this far up let's start maybe 8.0 1 let's delete some of these okay second grass  it's your table okay we're starting at Point 1 let's go down all the way and let's take a look at where we zero there it is okay now let's make that kind of the middle of our table so we see what's happening from both directions  alright so our values get closer and closer and closer to zero here they are there decreasing toward zero we can see we get real really really really really close to 2  and in the other direction as our values increase towards zero is be coming from we can see that our values and can get very very very very close to 2:16 okay next on note for ourselves you can stay numerically we're finding that  it approaches  two interesting now let's take a look at our grass so we're going to press  we already have it in our why you call so we can just press graph we should be good to go call Rec now this thing we can see as we come in from the left right as he's go to the increasing values it looks like you get very very very very close to 2  Emily come in from left we can see that we get very very very very very close to 2 zoom in and we're going to go right to the spot you want to see better Trace up there okay answer once you got your spot America little bit easier to see  so again as we get close to Zero from the right from the left it looks like our grass is approaching the value to so on our paper we can just make a note to myself also  graphically we can even show this one by sketching are graphs let's do that we had our point you hear we had our one or one or two let's make sure we Mark our coordinate plane as well  here is our and it kind of does this goes through 2 and continues to get larger continuous entry  okay and again this illustrates where our grass is going good it's getting constitute from both sides this function approaches to as X approaches 0 so what we get to State then in that the limit fast approaching zero for the function X over square root of x + 1 - 1 is equal to 2 okay  my next thing we're going to go over finding a limits Earl Ray what we're going to go what this is the key Concepts oh  the example I'd like to show you here is  fine the limit as X approaches two where FXR function Peace by and it is equal to 1 when X is not equal to 2 + 0 when x equals 2  okay if you having a hard time visualizing that lets draw grass real quick to make this a little bit easier so we're going to take graphical approach  sketch a graph sketch a graph we know we have have the one and the two at least on here on her x-axis and let's have R1 and R2 on the y-axis okay you label your numbers as well so  autograph is 1 whenever X is not too so that's everything every accident you except for two is going to be one we're going to draw that horizontal line heard me the horizontal line at y equals 1 but then we're going to leave a hole at 2 and we're going to keep going in the other direction enter natural shows that X is equal to earn the function is equal to 1 when X is everything except for two and then we need to show that when X is to our graph equals zero so we're going to make a closed Circle here at 2 and now we're going to find the limit as X approaches to so as our function approaches to we're going to find what value it and if we do that we take a look  we go in from the right we see that it approaches if we go in from the left we see that it approaches to as well and again you're seeing this pattern especially from the last example that what is the direction from the right and the direction from the left come to the same value then ding ding ding that's are so what we're going to say is that the limit as X approaches to for the function f of x equals 1 when X is not equal 2 + 0 + x equals 2 is equal to the value 1  now this is an important concept to go over because what about the sky right what about the fact that we know the function is actually two when we approached you fountains actually right here well here's the thing we were talking about limit that doesn't matter because it's important to know we can say that effort to equal 0 is irrelevant  and we can stay because a limit and what Xbox value a grass a part of me goodness gracious is what malavase ocean approaches from both approaches being the key word here it isn't relevant that are function Ashley equals we don't care about that we care about is what are function approaches and it approaches to so our limit is approaching one door limit is one okay sorry for all these little hiccups we got one more example let's get through it okay I will last topic we need to cover is when limits fail to exist so we're going to write that topic right here when limits fail to exist alright so the example we're going to do for this one  S2 show that the limit as X approaches 0 for absolute value X over X does not exist and we can just take the graphical approach here a graphically and you can throw this in your house Scotch the quick graph here if we were to graph the function absolute value of x over X I would get with you the following we have  play zoomed in all the way  we would get negative one one and I'm here we could have one up here we could have positive left us what would happen if you have open circles here and here and not make sense because we can't have zero in the denominator of a fraction  and then we would have Arrow  going to the right and arrows  going to the left it's important to remember that limit exists again only when we go in from the right and we got an where am I going from the left and we get a number and those numbers are the same just looking at this we can see that when we come in from the left  we get our grass approaches the value one when they come in from the right however we negative  I'm giving us values aren't the same we know that the limit does not exist and here's a few different ways to show that in your book  what we call the values really close to zero or really close to the on either side are called Delta so with label those it looks almost like a funny-shaped s and all that Omni inter negative Delta to zero an extra saying all these really really small values that could close to zero and not little range right there Cinemark at lemons for only interested as they get very very close very very close area we found the x equals negative one and the other really really close to area from 0 to positive Delta Lisa x equals 1 powder  and yet another way to write this information what we just found it to write out that the limits as X approaches 0 from the right the negative Direction equals 1 limit as X approaches 0 from the left is equal to positive one negative friend me  negative and positive one so as you can see these two values are different and when we have the directional approaching approaching different numbers that we don't have a limit so we can stay  the limit as X approaches 0 for absolute value X over X is non elephant  is 9% or does not exist okay moving on last time I slept over a page my notes Here one second okay last topic on founded behave unbounded Behavior and here's an example of this circumstance discuss the existence the limit as X approaches 0 for the function 1 over x squared  again we have our two approaches we have our numeric approach so let's take a look at that let's look in our calculator  Escobar y equals how much time in this function sing by X  squared  and then let's make sure our tables that is good to go hopefully get this one right on the 1st why is timeless see how we could do alright we want to get really close to zero so it looks like write a good place to be doing that and let's take a look at our table  okay so zeros right in the middle of their that's exactly what we want to see and know we want to take a look at the values as the approach zero so  what we're seeing here is something very strange we what is really big numbers that just keep on getting bigger until we get the thing that we can't be they keep on getting larger and larger and larger until we get to that value undefined  here's the thing you can kind of ask yourself here do they approach just one value like all our other limits hat and the answer is no they don't they approached ever-increasing values which is the nature of exponential function write something squared is it going to Forever increase exponentially so it's not going to ever touch the asymptote at zero it's just going to keep on increasing an increasing forever as we get closer and closer to 0 so you're looking at our values and that makes sense that they aren't going to touch zero and they're not going to approach the value let's draw the graph as well just to be sure okay always got to plot your points here let's just label a few of them so we have a scale going on that make sense okay so if you graph this in your calculator again we can just do a quick preview what that would look like if you press crafts my goodness is Brighton okay we were pretty good illustration of what that looks like we've got our prayers going up from both directions and give me have that ass until 20 we know we can't be zero because of the type undefined at the bottom of a fraction you know that that's not cool to have  and again curve at from both directions  cat set can see if we do our traditional limit thing where we see what it approaches we follow the graph for both as we run into the same problem that from the new Michael approach for never our fingers are never going to touch up something right there never going to meet out of number that's the same because these funds are never going to stop going up so what we can State then is  because after that not become closest two a single number as ax approaches zero the limit Adnan eggs distance non-existent okay that's the last thing I had to show your face for watching will see you in class tomorrow

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AP Calc 1.2a

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