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Let's try to come up with a mathematically rigorous definition for what this statement means.

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The statement that f(x) as x approaches c is equal to L

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So let's say that this means that you can get

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you can get f(x)

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as close to L as you want

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I'll put that in quotes right over here...

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but as close as you want

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by getting x sufficiently close

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to c, so another way of saying this is if you tell me

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"hey I want to get my f(x) to be within 0.5 of this limit", then you're telling me if this

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limit is actually true, then you should be able to hand me a value around

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c that if x is within that range, that f(x) is definitely going to be as close to L as I desire.

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So let me draw that out to make it a little bit clearer and I'm gonna zoom in

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and I'll draw another diagram. So let's say that...

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that is my y-axis and I'm gonna zoom in

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and I'll draw a slightly different function just so that we can

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focus on what's going on around here, the range

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is around c and the range is around L

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so that's x

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right over here is y

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this is c, and let's just say

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our function is doing something like

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doing something like --let's see I don't want it to be defined

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at c, at least just for the--. It could be, you could always find the limit even

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when it's undefined. But let's just say our function looks something like that.