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Current time:0:00Total duration:3:36

Formal definition of limits Part 1: intuition review

Video transcript

let's review our intuition of what a limit even is so let me draw some axes here so let's say this is my y-axis so try to draw a vertical line so that right over there is my y-axis and then let's say this is my x-axis I'll focus on the first quadrant although I don't have to so let's say this right over here is my x-axis and then let me draw a function so let's say my function looks something like that it could look like anything but that seems suitable so this is a function y is equal to f of X and just for the sake of conceptual understanding I'm going to say it's not defined at a point I didn't have to do this you can find the limit as X approaches a point where the function actually is defined but it becomes that much more interesting at least for me or you start to understand why a limit might be relevant where a function is not defined at some point so the way I've drawn it this function is not defined this function is not defined when X is equal to is when X is equal to C now the way that we thought about a limit is what does f of X approach as X approaches C so let's think about that a little bit when X is a reasonable bit lower than C f of X for our function that we just drew is right over here that's what f of X is going to be equal Y is equal to f of X when X gets a little bit closer then our f of X is right over is right over there when X gets even closer maybe really almost at C but not quite C then our f of X is right over here and the way we see it we see that our f of X of seems to be as we get as X gets closer and closer to C as X gets closer and closer to C it looks like our f of X is getting closer and closer to some value it's getting closer and closer to some value right over there I'll even draw it maybe I'll draw it with a more solid line and that was actually only the case when X was getting closer to C from the left from values of X less than C but what happens as we get closer to get closer and closer to cease from values of X that are larger than C well when X is over here of X is right over here and so that's what f of X is is right over there when X gets a little bit closer to C our f of X is right over there when f of when X is just very slightly larger than C then our f of X is right over there and you see once again it seems to be approaching it seems to be approaching that same value and we call that value that value that f of X seems to be approaching as X approaches C we call that value L or the limit and so the way we would denote it is we would call that the limit it doesn't have to call it L all the time but it's referred to as the limit and the way that we would kind of write that mathematically is we would say the limit the limit of f of X as X approaches C is equal to is equal to L and this is a fine conceptual understanding of limits and it really will take you pretty far and you're ready to progress and start thinking about taking a lot of limits but this isn't a very mathematically rigorous definition of limits and so this sets us up for the intuition in the next few videos we will introduce a mathematically rigorous definition of limits that will allow us to do things like prove that the limit as X approaches C truly is equal to L
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