Introduction to limits (old)

welcome to the presentation on limits let's get started with some well first an explanation before I do any problems so let's say I had let me make sure I have the right color and my pen works okay let's say I had the limit and I'll explain what a limit is in a second but the way you write it is hey you say what's the limit oh and my color I'm the wrong okay I need to use a pen I'm yellow okay the limit as X approaches 2 of x squared of x squared now all this is saying is what value does this expression x squared approach as X approaches 2 well I this is pretty easy if we look at it let me draw draw a graph stay in this yellow color so let me draw x squared x squared looks something like let me use a different color let's x squared looks something like this right alright and when X is equal to 2 when X is equal to 2 y or the expression because we don't say what this is equal to is just the expression x squared is equal to 4 right it's like that right so a limit is saying as X approaches 2 and as X approaches 2 from both sides from the from numbers less than 2 and from numbers right than 2 what does the expression approach so and and you might I think already see where this is going and be wondering why we're even going to the trouble of learning this new concept cuz it seems pretty obvious but as X as we get to X is closer and closer to 2 from this direction and as we get to X is closer and closer to this direction what does this expression equal well it essentially equals 4 right the expression is equal to or the way I think about it is as you move on the curve closer and closer to the expressions value what does the expression equal in this case it equals four so you're probably saying Sal this seems like a useless concept because I could have just stuck two in there and I know that you know if this is say this is f of X that if f of X is equal to x squared that F of two is equal to four and that would have been a no-brainer well let me let me maybe give you one wrinkle on that and then hopefully now you'll start to see what the use of a limit is let me define let me say f of X f of X is equal to x squared when if X does not equal two and let's say it equals three when x equals two interesting so there's a slight variation on on this expression right here so this is our new f of X so let me ask you a question what is my pinch don't works what is the limit I used courtesy of this time what is the limit as a as X that's an X as X approaches 2 of f of X that's an X that says X approaches 2 it's just like that I just I don't know for some reason my brain isn't working functionally okay so let me let me let me graph this now so as an equally neat looking graph that's the one I just drew let me draw so now it's almost the same as this curve except something interesting happens at x equals two so it's just like this it's like an x squared curve like like that but at x equals two and f of X equals four we draw a little hole we draw a hole right because it's not defined at x equals 2 this is x equals 2 this is 2 this is 4 this is the f of x axis of course and when x is equal to 2 let's say this is 3 when X is equal to 2 f of X is equal to 3 this is actually right below this I should it doesn't look completely blow it but I think you you got to get the picture see this graph is x-squared it's exactly x-squared until we get to x equals 2 at x equals 2 we have a graph kinetograph we have a gap in the graph which maybe could be called a grap we have a gap in the graph and then we keep and then after x equals 2 we keep moving on and that gap and that gap is defined right here what happens when x equals 2 well then f of X is equal to 3 so this graph kind of goes it's just like x squared and then it's at but instead of F of 2 being for F of 2 drops down to 3 but then we keep on going so going back to the limit problem what is the limit as X approaches 2 now well let's think about the same thing we're going to go that this is how I visualize it I go along the curve so let me pick a different color so as X approaches 2 from this side from the left-hand side or from numbers less than 2 f of X is approaching values approaching 4 right f of X is approaching 4 as you go as X approaches 2 right I think you see that if you just follow along the curve as you approach F of 2 you get closer and closer to 4 similarly as you go from the right-hand side and make sure my things still working as you go from the right-hand side you go along the curve and f of X is also slowly approaching 4 so as you can see as we go closer and closer closer to x equals to f of whatever number that is approaches 4 right so in this case the limit as X approaches 2 is also equal to is equal to 4 well this is interesting because in this case the limit the limit as X approaches 2 of f of X does not equal F of 2 right now normally this would be on this line this is in in this case the limit as you approach the question is equal to evaluating the expression of that value in this case the limit isn't so this is I think now you're trying to see why the limit is a slightly different concept than just evaluating the function at that point because you have functions where for whatever reason at a certain point either the function might not be defined or the function kind of jumps up or down but as you approach that point you still approach a value different than the function at that point now that's my introduction I think this will give you an intuition for what a limit is in another presentation I'll give you the more formal mathematical you know the Delta Epsilon a definition of a limit and actually in the very next module I'm not going to do a bunch of problems involving the limit I think as you do more and more problems you'll get more and more of an intuition as to what a limit is and then as we go into derivatives and integrals you'll actually understand why why people probably even invented limits to begin with we'll see you in the next presentation