Epsilon Delta Limit Definition 1 Introduction to the Epsilon Delta Definition of a Limit.
Epsilon Delta Limit Definition 1
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- Let me draw a function that would be interesting
- to take a limit of.
- And I'll just draw it visually for now, and we'll do some
- specific examples a little later.
- So that's my y-axis, and that's my x-axis.
- And let;s say the function looks something like--
- I'll make it a fairly straightforward function
- --let's say it's a line, for the most part.
- Let's say it looks just like, accept it has a
- hole at some point.
- x is equal to a, so it's undefined there.
- Let me black that point out so you can see that
- it's not defined there.
- And that point there is x is equal to a.
- This is the x-axis, this is the y is equal f of x-axis.
- Let's just say that's the y-axis.
- And let's say that this is f of x, or this is
- y is equal to f of x.
- Now we've done a bunch of videos on limits.
- I think you have an intuition on this.
- If I were to say what is the limit as x approaches a,
- and let's say that this point right here is l.
- We know from our previous videos that-- well first of all
- I could write it down --the limit as x approaches
- a of f of x.
- What this means intuitively is as we approach a from either
- side, as we approach it from that side, what does
- f of x approach?
- So when x is here, f of x is here.
- When x is here, f of x is there.
- And we see that it's approaching this l right there.
- And when we approach a from that side-- and we've done
- limits where you approach from only the left or right side,
- but to actually have a limit it has to approach the same thing
- from the positive direction and the negative direction --but as
- you go from there, if you pick this x, then this is f of x.
- f of x is right there.
- If x gets here then it goes here, and as we get closer and
- closer to a, f of x approaches this point l, or this value l.
- So we say that the limit of f of x ax x approaches
- a is equal to l.
- I think we have that intuition.
- But this was not very, it's actually not rigorous at all
- in terms of being specific in terms of what we
- mean is a limit.
- All I said so far is as we get closer, what does
- f of x get closer to?
- So in this video I'll attempt to explain to you a definition
- of a limit that has a little bit more, or actually a lot
- more, mathematical rigor than just saying you know, as x gets
- closer to this value, what does f of x get closer to?
- And the way I think about it's: kind of like a little game.
- The definition is, this statement right here means that
- I can always give you a range about this point-- and when I
- talk about range I'm not talking about it in the whole
- domain range aspect, I'm just talking about a range like you
- know, I can give you a distance from a as long as I'm no
- further than that, I can guarantee you that f of x is go
- it not going to be any further than a given distance from l
- --and the way I think about it is, it could be viewed
- as a little game.
- Let's say you say, OK Sal, I don't believe you.
- I want to see you know, whether f of x can get within 0.5 of l.
- So let's say you give me 0.5 and you say Sal, by this
- definition you should always be able to give me a range
- around a that will get f of x within 0.5 of l, right?
- So the values of f of x are always going to be right in
- this range, right there.
- And as long as I'm in that range around a, as long as I'm
- the range around you give me, f of x will always be at least
- that close to our limit point.
- Let me draw it a little bit bigger, just because I think
- I'm just overwriting the same diagram over and over again.
- So let's say that this is f of x, this is the hole point.
- There doesn't have to be a hole there; the limit could equal
- actually a value of the function, but the limit is more
- interesting when the function isn't defined there
- but the limit is.
- So this point right here-- that is, let me draw the axes again.
- So that's x-axis, y-axis x, y, this is the limit point
- l, this is the point a.
- So the definition of the limit, and I'll go back to this in
- second because now that it's bigger I want explain it again.
- It says this means-- and this is the epsilon delta definition
- of limits, and we'll touch on epsilon and delta in a second,
- is I can guarantee you that f of x, you give me any
- distance from l you want.
- And actually let's call that epsilon.
- And let's just hit on the definition right
- from the get go.
- So you say I want to be no more than epsilon away from l.
- And epsilon can just be any number greater, any real
- number, greater than 0.
- So that would be, this distance right here is epsilon.
- This distance there is epsilon.
- And for any epsilon you give me, any real number-- so this
- is, this would be l plus epsilon right here, this would
- be l minus epsilon right here --the epsilon delta definition
- of this says that no matter what epsilon one you give me, I
- can always specify a distance around a.
- And I'll call that delta.
- I can always specify a distance around a.
- So let's say this is delta less than a, and this
- is delta more than a.
- This is the letter delta.
- Where as long as you pick an x that's within a plus delta and
- a minus delta, as long as the x is within here, I can guarantee
- you that the f of x, the corresponding f of x is going
- to be within your range.
- And if you think about it this makes sense right?
- It's essentially saying, I can get you as close as you want to
- this limit point just by-- and when I say as close as you
- want, you define what you want by giving me an epsilon; on
- it's a little bit of a game --and I can get you as close as
- you want to that limit point by giving you a range around the
- point that x is approaching.
- And as long as you pick an x value that's within this range
- around a, long as you pick an x value around there, I can
- guarantee you that f of x will be within the range
- you specify.
- Just make this a little bit more concrete, let's say you
- say, I want f of x to be within 0.5-- let's just you know, make
- everything concrete numbers.
- Let's say this is the number 2 and let's say this is number 1.
- So we're saying that the limit as x approaches 1 of f of x-- I
- haven't defined f of x, but it looks like a line with the hole
- right there, is equal to 2.
- This means that you can give me any number.
- Let's say you want to try it out for a couple of examples.
- Let's say you say I want f of x to be within point-- let me do
- a different color --I want f of x to be within 0.5 of 2.
- I want f of x to be between 2.5 and 1.5.
- Then I could say, OK, as long as you pick an x within-- I
- don't know, it could be arbitrarily close but as long
- as you pick an x that's --let's say it works for this function
- that's between, I don't know, 0.9 and 1.1.
- So in this case the delta from our limit point is only 0.1.
- As long as you pick an x that's within 0.1 of this point, or 1,
- I can guarantee you that your f of x is going to
- lie in that range.
- So hopefully you get a little bit of a sense of that.
- Let me define that with the actual epsilon delta, and this
- is what you'll actually see in your mat textbook, and then
- we'll do a couple of examples.
- And just to be clear, that was just a specific example.
- You gave me one epsilon and I gave you a delta that worked.
- But by definition if this is true, or if someone writes
- this, they're saying it doesn't just work for one specific
- instance, it works for any number you give me.
- You can say I want to be within one millionth of, you know, or
- ten to the negative hundredth power of 2, you know, super
- close to 2, and I can always give you a range around this
- point where as long as you pick an x in that range, f of x will
- always be within this range that you specify, within that
- were you know, one trillionth of a unit away from
- the limit point.
- And of course, the one thing I can't guarantee is what
- happens when x is equal to a.
- I'm just saying as long as you pick an x that's within my
- range but not on a, it'll work.
- Your f of x will show up to be within the range you specify.
- And just to make the math clear-- because I've been
- speaking only in words so far --and this is what we see the
- textbook: it says look, you give me any epsilon
- greater than 0.
- Anyway, this is a definition, right?
- If someone writes this they mean that you can give them any
- epsilon greater than 0, and then they'll give you a delta--
- remember your epsilon is how close you want f of x to be
- to your limit point, right?
- It's a range around f of x --they'll give you a delta
- which is a range around a, right?
- Let me write this.
- So limit as approaches a of f of x is equal to l.
- So they'll give you a delta where as long as x is no more
- than delta-- So the distance between x and a, so if we pick
- an x here-- let me do another color --if we pick an x here,
- the distance between that value and a, as long as one, that's
- greater than 0 so that x doesn't show up on top of a,
- because its function might be undefined at that point.
- But as long as the distance between x and a is greater
- than 0 and less than this x range that they gave you,
- it's less than delta.
- So as long as you take an x, you know if I were to zoom the
- x-axis right here-- this is a and so this distance right here
- would be delta, and this distance right here would be
- delta --as long as you pick an x value that falls here-- so as
- long as you pick that x value or this x value or this x value
- --as long as you pick one of those x values, I can guarantee
- you that the distance between your function and the limit
- point, so the distance between you know, when you take one of
- these x values and you evaluate f of x at that point, that the
- distance between that f of x and the limit point is
- going to be less than the number you gave them.
- And if you think of, it seems very complicated, and I have
- mixed feelings about where this is included in most
- calculus curriculums.
- It's included in like the, you know, the third week before you
- even learn derivatives, and it's kind of this very mathy
- and rigorous thing to think about, and you know, it tends
- to derail a lot of students and a lot of people I don't think
- get a lot of the intuition behind it, but it is
- mathematically rigorous.
- And I think it is very valuable once you study you know, more
- advanced calculus or become a math major.
- But with that said, this does make a lot of sense
- intuitively, right?
- Because before we were talking about, look you know, I can get
- you as close as x approaches this value f of x is going
- to approach this value.
- And the way we mathematically define it is, you say Sal,
- I want to be super close.
- I want the distance to be f of x [UNINTELLIGIBLE].
- And I want it to be 0.000000001, then I can always
- give you a distance around x where this will be true.
- And I'm all out of time in this video.
- In the next video I'll do some examples where I prove the
- limits, where I prove some limit statements using
- this definition.
- And hopefully you know, when we use some tangible numbers, this
- definition will make a little bit more sense.
- See you in the next video.
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