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Let's review our intuition of what a limit even is.

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So, let me draw some axes here.

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Let's say this is my y axis, so try to draw my vertical line

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That right over there is my y axis

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and let's say this is my x axis.

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I'll focus on the first quadrant, although I don't have to...

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Let's say this right over here is my x axis and let me draw a function.

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So let's say my function looks something like that.

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It could look like anything suitable, so this is y is equal to f of x

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and just for the sake of conceptual understanding

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I'm going to say it's not defined at a point.

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I didn't have to do this, you could find the limit as X approaches a point

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where the function actually is defined,

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but it becomes that much more interesting, at least for me

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where you start to understand where a limit might be relevant

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when a function is not defined at some point.

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So, the way I've drawn it this function is not defined

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when x is equal to c

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Now, the way that we've though about a limit is,

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what does f(x) approach as x approaches c?

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So, let's think about that a little bit.

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When x is a reasonable bit lower than c,

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f(x) for our function that we just drew is right over here

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y is equal to f(x)

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when x gets a little bit closer, then our f(x) is right over there

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when x gets even closer; It's really almost at c but not quite c

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then our f(x) is right over here

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and the way we see it,

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we see that our f(x) seems to be that as x gets closer to c it looks like our f(x) is getting closer to some value.

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right over there, I'll even draw it with a more solid line.

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and that was only the case

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when x was getting closer to c from the left,

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from values less than c.

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But what happens when we get closer and closer to c

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from values of x that are larger than c,

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well when x is over here, f(x) is over here.

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That's what f(x) is

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when x is over there, x gets a little bit closer to c

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our f(x) is right over there,

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when x is just very slightly larger than c,

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then our f(x) is right over there

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and you see once again it seems to be approaching

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that same value

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and we call that value, the value f(x) seems to be approaching as x approaches c,

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we call that value L, or the limit

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and we'd call the limit, we don't have to call it L all the time.

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and the way that we would write that mathematically is,

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the limit of f(x) as x approaches c is equal to L

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and this is a fine conceptual understanding of limits

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and it really will take you very far,

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and you are ready to progress and start taking a lot of limits.

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But this isn't a very mathematically rigourous definition of limits.

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This sets us up for the intuition,

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in the next few videos we will introduce

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a mathematically rigourous definition of limits.

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That will allow us to do things.

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Like prove that the limit as x approaches c

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truly,

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is equal to L.