in chapter 2 we begin our study of calculus with the concept of the limit and as we'll come to see the limit is fundamental to calculus and in my opinion it's not an overstatement to say that calculus does not exist without the limit i later on this semester we'll be developing two very important tools those tools being called the derivative and the integral and what we'll see is that both of those tools trace back to this concept of the limit so the limit is very important and it is foundational to everything we do in calculus and so in chapter 2 we're going to have a pretty intense focus on limits and learning a lot of things about limits and how to work with them later on in chapter 2 we'll define a limit formally but for now we're going to take a an informal approach so in section 2.1 we're just going to really discuss very very informally the idea of limits so i like to start this with the concept of a function so we talk about functions quite frequently in mathematics so the idea of a function is you put in an input and you get out an output so for instance if we have a function f and that function has some point in its domain called x equals c then what we can do is we can calculate the output by calculating this number f of c so input c output f of c so for instance if f of x is equal to 3x minus 1 then f of 2 is going to come out to be 3 times 2 minus 1 which is of course 5. so very easily i'm able to calculate the input excuse me i'm very easy very easily able to calculate the output from an input but there are functions where the input does not readily give me an output so for instance let's think about this function here the function f of x equals x squared minus 9 divided by x minus 3. so you'll notice that 3 is not in the domain of this function so first off you see that 3 minus 3 on the bottom here would be 0 and of course division by 0 is undefined and so if i want to calculate f of 3 i can't i cannot possibly do that in particular here when i calculate f of 3 i end up with this 0 over 0. now this is a an important expression here in calculus this is called an indeterminate form so you want to be very careful 0 over 0 is not equal to division by zero is never allowed so this expression here is not equal to one we call this an indeterminate form uh and later on in the semester we'll talk about other indeterminate forms but for now right here zero over zero is the main indeterminate form that we'll look at in this in this particular section so very very informally what we might say is that the limit is what f 3 should be now notice i can't find f of 3 directly because plugging in 3 i have this indeterminate form which is not a number it's simply an expression so the indeterminant form doesn't tell me anything about what f of 3 should be so i i'm not able to just calculate the output from the input so the question here is what what if anything should we define as the this quantity of what f 3 should be so what we'll be able to show later on is this expression here so the limit again is what f of 3 quote unquote should be and we're going to say that this f of 3 should be equal to 6 very very loose terms here so we're not really talking about anything in formal formal mathematical terms so we write this expression the limit as x approaches 3 of x squared minus 9 divided by x minus 3 is equal to 6. so very soon we will be able to show that exactly so again just a quick recap here i cannot find f of 3 so this idea of the limit is saying what should f 3 be equal to and so here we write this new expression we'll be using this notation quite often this limit as x approaches 3 of our function is equal to 6. so let's explore this idea of a limit through the concept of velocity in particular average velocity so the average velocity is defined this way it's the displacement divided by the time elapsed so on some interval time sometime interval t1 t2 our displacement over time elapsed we'll write as delta s over delta t which has this form here so it's just a change in position which is called displacement over the change in time so that's our average velocity so very simply the position of a rock uh t seconds after being launched at a speed of 96 feet per second is given by this function s of t equals negative 16 t squared plus 96 t feet so the average velocity between time 1 second and time 3 seconds is simply going to be this expression s 3 minus s of 1 divided by 3 minus 1. so these are function evaluations of this function f and what we'll find is we have a average velocity of 32 feet per second now geometrically what this is giving us is the slope of a line so in particular it's giving us the slope of the line passing through these two points on our position curve so time 1 we have a displacement of 80 feet times two we have a excuse me time three we have a displacement of 144 feet so this change in position which is our displacement right that's on the y-axis here divided by our change in time that's on the x-axis the slope of this line is that 32 we that we just calculated so a line passing through two points on a curve like this is called a secant line so our geometric interpretation of average velocity it's the slope of the secant line so the average rate of change in general is defined this way if i have just a general function on a closed interval then this quantity f of b minus f of a divided by b minus a is called the average rate of change uh in particular the average rate of change of a position function is called average velocity but if our function has no particular context or a unique context removed from the context of position or motion we'll simply call it the average rate of change and what you should notice here is this is just the slope of a line so this is like y2 minus y1 divided by x2 minus x1 so let's take this idea of average velocity and move over towards a more sophisticated idea of instantaneous velocity or instantaneous rate of change so i'm not interested in the the velocity on a time interval like between time one and time three i'm interested in the time the velocity at a particular time say for instance four seconds or the velocity at 3 seconds so the instantaneous velocity how might we approach that well we might think to just apply the average velocity formula but the problem here is that we're only looking at a particular time point so very naively if i approach this two if i approach this too loosely i might think to just do this here but very quickly we would notice that the top is zero and the bottom is zero so here we are again zero over 0 is indeterminate so let's see if we can figure out what the instantaneous velocity should be at some particular time so again we're going to tie this back to this idea of a limit so let's suppose we want the instantaneous velocity at time equals one second after the rock was launched so we're going to calculate the average velocity on some different time intervals so the left hand time is time one that's the time we're interested in and what we're going to do is we're going to take this t1 we're going to pick some values for it and push those values back towards one so let's see the first interval let's say the time interval one to one and a half seconds we make our calculation and we have 56 so this is going back to that function that we just the position function of the rock being launched that we were just talking about in the previous example so the next time interval we'll look at the time interval 1 to 1.1 seconds we'll calculate the average velocity and we have 62.4 right now notice this right hand endpoint of this time interval is getting closer and closer to 1. so here we are i've picked the time interval 1 to 1.01 seconds we calculate the average velocity and we have 63.84 and then we continue that pattern the time interval 1 to 1.001 seconds our average velocity is 63.984 so perhaps you're noticing a very clear trend here as this t1 approaches one so this notation here this we read this as t one approaches one so t one is this second number as this second number is getting closer and closer to one does it seem like our average velocity is arriving at a destination and so very very hopefully very obviously it does seem to you that this is approaching the value of 64. so notice the jumps start to get smaller and we seem to be settling on a particular value so from this table of values here it seems reasonable to say that our instantaneous velocity is going to be 64 feet per second so we'll write this the instantaneous velocity is the limit as time one approaches one of our expression for the average velocity and we're going to call that instantaneous velocity 64 feet per second so as this time interval shrinks we get some nice behavior in this average velocity and it seems to me and hopefully to you that these numbers here are approaching 64. so as i mentioned as you'll soon come to appreciate every fundamental task in calculus involves the calculation of a limit we'll get to a point where we won't necessarily be observing a limit in everything that we do uh we will will ultimately want to be work smarter and work uh more efficiently uh so when we get to derivatives and when we get to the integral again everything under the hood there is going to come back to a limit um we won't always be writing these things as their limit definitions but if we peel everything away everything pretty much everything we do involves the calculation of a limit this will bring me to the the idea of a tangent line so let's let's see if we can make a geometric connection between the average and the instantaneous velocity so a tangent line perhaps you're familiar with the idea of a tangent line so a tangent line to a curve at a point x y is a line that just touches the curve at that point and also has the same direction as the curve all right so a tangent line just sort of glances by a curve at a particular point touches and and moves on so let's see if we can determine a geometric connection between the average and the instantaneous velocity so in this picture here i have the position function for that rock that was launched so the the line in black is the secant line so the secant line is passing through time one and time two so again the black line here the slope of the secant line is the average velocity so the slope of this black line here is 48. all right delta t here represents the spacing between the two time points so time one to time two that's delta t of one and then as we showed the instantaneous velocity at time one was 64 feet per second so in red here the line in red is the tangent line so let's see if we can make a connection between the secant line and the tangent line as this delta t gets forced down to zero in other words these two black points are going to be pushed closer and closer and closer to each other so here's delta t equals one the black line is the secant the red line is the tangent next up is delta t equals a half so the slope of the black line is 56 next up is delta t of 0.1 the slope of the black line is 62.4 and maybe you're starting to see what's happening next up delta t 0.01 at this point the black line and the red line are basically indistinguishable on this graphic and we'll go one more step delta t equals point zero zero one so again those two these two points here are extremely close to each other it looks like one point but there are in fact two and we're noticing the secant line has a slope of 63.984 we were we determined that the instantaneous velocity was 64. so let's watch this one more time back from the beginning so here's delta t equals one the black line is the secant that's my average velocity and the red line is the tangent so as delta t is getting smaller and approaching zero the black line the tangent line is approaching the red line which is the secant excuse me which is the tangent line so one more time delta t equals one black is secant red is tangent as delta t goes to zero the secant is approaching the tangent all right so we might now say that the slope of the tangent line is our instantaneous velocity so we've shown here just in a graphical way that as the time interval shrinks to zero the tangent line and the secant line are approaching so the secant average of secant is our average velocity slope 63.984 tangent has our instantaneous velocity with slope 64. so we might now say that the instantaneous velocity or the instantaneous rate of change we're going to make the connection that the instantaneous velocity or the instantaneous rate of change is the slope of the tangent line so let's make some conclusions here so very informally we talked about a limit we talked about a limit as being the sort of expected output of some function or expression call that number l which we can't immediately calculate and we introduce this new notation the limit as x goes to c of f of x is equal to l in terms of this idea of velocity we have determined that the instantaneous velocity is the limit of the average velocities as we push the width of our time interval to zero so two ways of expressing that the limit as t2 approaches t1 of our average velocity or we might say as or we might say the limit as delta t approaches zero of displacement over time we have our geometric connection as well geometrically the average velocity is the slope of the secant line passing through two points call those points p and q while the instantaneous velocity is the slope of the line excuse me a slope of the tangent line passing through that single point p so this calculation of the slope of a tangent line is not possible through standard algebraic means if you think about calculating the slope of a line you need two points a tangent line you only have one point so if i'm going to calculate a slope of a tangent line that's not going to be possible because i'm going to require two particular points and a tangent line is only going to guarantee me one so calculating the slope of the tangent line or i might say calculating the instantaneous rate of change or i might say calculating the average or the instantaneous velocity that's going to require calculus and that's going to require this concept of a limit so in this section we're just very informally introducing this idea of a limit so in the coming sections we'll be learning some limit theorems how do we work how to work with limits uh uh graphically how to work with limits algebraically um and going through some different styles of limits and then in chapter three we developed we developed the derivative which we will see is itself defined as a particular limit as a reminder some of these things might be might be needed throughout this course um in particular the slope of the excuse me in particular the equation of a line in point-slope form is something that we'll be using quite frequently uh in chapters two and possibly chapter three uh so we will often write the equation of a tangent line and it's helpful to know point-slope form here