Limit Examples (part 2) More limit examples
Limit Examples (part 2)
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- OK, hopefully, my tool is working now.
- But anyway, so we were saying when x is equal to minus 0.001,
- so we're getting closer and closer to 0 from the negative
- side, f of x is equal to minus 1,000, right?
- You can just evaluate it yourself, right?
- And as you see, as x approaches 0 from the negative direction,
- we get larger and larger-- or I guess you could say smaller and
- smaller negative numbers, right?
- You get-- you know, if it's minus 0.0001, you'd get minus
- 10,000, and then minus 100,000, and then minus 1 million, you
- could imagine the closer and closer you get to zero.
- Similarly, when you go from the other direction, when you say
- what is-- when x is 0.01, there you get positive 100, right?
- When x is point-- the thing is frozen again-- when it's 0.001,
- you get positive 1,000.
- So as you see, as you approach 0 from the negative direction,
- you get larger and larger negative values, or I guess
- smaller and smaller negative values.
- And as you go from the positive direction, you get larger
- and larger values.
- Let me graph this just to give you a sense of what this graph
- looks like because this is actually a good graph to know
- what it looks like just generally.
- So let's say I have the x-axis.
- This is the y-axis.
- Change my color.
- So when x is a negative number, as x gets really, really,
- really negative, as x is like negative infinity, this
- is approaching zero, but it's still going to be a
- slightly negative number.
- And then as we see from what we drew, as we approach x is equal
- to 0, we asymptote, and we approach negative
- infinity, right?
- And similarly, from positive numbers, if you go out to
- the right really far, it approaches 0, but it's
- still going to be positive.
- And as we gets closer and closer to 0, it spikes up, and
- it goes to positive infinity.
- You never quite get x is equal to 0.
- So in this situation, you actually have as x approaches--
- so let me give you a different notation, which you'll
- probably see eventually.
- I might actually do a separate presentation on this.
- The limit as x approaches 0 from the positive direction,
- that's this notation here, of 1/x, right?
- So this is as x approaches 0 from the positive direction,
- from the right-hand side, well, this is equal to infinity.
- And then the limit as x-- this pen, this pen-- the limit as x
- approaches 0 from the negative side of 1/x.
- This notation just says the limit as I approach
- from the negative side.
- So as I approach x equal 0 from this direction, right, from
- this direction, what happens?
- Well, that is equal to minus infinity.
- So since I'm approaching a different value when I
- approach from one side or the other, this limit
- is actually undefined.
- I mean, we could say that from the positive side, it's
- positive infinity, or from the negative side, it's negative
- infinity, but they have to equal the same thing for
- this limit to be defined.
- So this is equal to undefined.
- So let's do another problem, and I think this should
- be interesting now.
- So let's say, just keeping that last problem we had in mind,
- what's the limit as x approaches 0 of 1/x squared?
- So in this situation, I'll draw the graph.
- That's my x-axis.
- That's my y-axis.
- So here, no matter what value we put into x, we get a
- positive value, right?
- Because you're going to square it.
- If you put minus-- you could actually-- oh, let me do it.
- It'll be instructive, I think.
- Once again, obviously you can't just put x equal to 0.
- You'll get 1/0, which is undefined.
- But let's say 1 over x squared.
- What does 1 over x squared evaluate to?
- So when x is 0.1, 0.1 squared is 0.01, so 1/x is 100.
- Similarly, if I do minus 0.1, minus 0.1 squared is positive
- 0.01, so then 1 over that is still 100, right?
- So regardless of whether we put a negative or positive number
- here, we get a positive value.
- And similarly, if I put-- if we say x is 0.01, if you evaluate
- it, you'll get 10,000, and if we put minus 0.01, you'll get
- positive 10,000 as well, right?
- Because we square it.
- So in this graph, if you were to draw it, and if you have a
- graphing calculator, you should experiment, it
- looks something like this.
- I can see this dark blue.
- So from the negative side, it approaches infinity, right?
- You can see that.
- As we get to smaller and smaller-- as we get closer and
- closer to 0 from the negative side, it approaches infinity.
- As we go from the positive side-- these are actually
- symmetric, although I didn't draw it that symmetric-- it
- also approaches infinity.
- So this is a case in which the limit-- oh, that's
- not too bright.
- I don't know if you can see -- the limit as x approaches 0
- from the negative side of 1 over x squared is equal to
- infinity, and the limit as x approaches 0 from the positive
- side of 1 over x squared is also equal to infinity.
- So when you go from the left-hand side, it
- equals infinity, right?
- It goes to infinity as you approach 0.
- And as you go from the right-hand side, it
- also goes to infinity.
- And so the limit in general is equal to infinity.
- And this is why I got excited when I first started
- learning limits.
- Because for the first time, infinity is a legitimate answer
- to your problem, which, I don't know, on some metaphysical
- level got me kind of excited.
- But anyway, I will do more problems in the next
- presentation because you can never do enough limit problems.
- And in a couple of presentations, I actually give
- you the formal, kind of rigorous mathematical
- definition of the limits.
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