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Studying for a test? Prepare with these 14 lessons on Limits and continuity.
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Video transcript
Let's say that f of x is equal to 1 over x. And we want to think about what the limit of f of x is as x approaches 0 from the positive direction. And to think about this I'm going to set up a little table here. So let's set up x and then let's think about what f of x is going to be. I'm going to approach 0 from the positive direction. So let's say we try 0.1. Then we're going to try 0.01. Then we can try 0.001. Then we can try 0.0001. So notice-- each of these numbers-- they're all larger than 0 and they're approaching 0 from the positive direction. We're getting closer and closer and closer to 0. So when x is 0.1, f of x is just going to be 1 over this. This is 1/10 so 1 over that is just going to be 10. To 1 over 0.01 is going to be 100. 1 over 0.001 is going to be 1,000. 1 over 0.0001 is going to be 10,000. So you see, as x gets closer and closer to 0 from the positive direction, f of x just really grows really, really fast. So what we say here is the limit of f of x as x approaches 0 from the positive direction is going to be equal to positive infinity. Or we could just write infinity. This thing over here-- if we put something really, really close, so if we say 0 point seven digits behind the decimal place, then 1 over that's going to be 1 with one, two, three, four, five, six, seven zeros. Did I do that right? Here I had four places behind the decimal, four 0's. Here I have one, two, three, four, five, six, seven, and here I have seven 0's. So you see, as we get closer and closer to 0 from the positive direction, the f of x just gets larger and larger and larger. It's just completely unbounded. So we'd say this is equal to infinity. Well, let's think about another limit. Let's think about the limit as x approaches 0 from the negative direction of f of x, or the limit of f of x as x approaches 0 from the negative direction. Well in that case, we can just make each of these values negative. So if x is negative 0.1 then this is going to be negative 10. If this is negative, then this is negative. If this is negative, then this is negative. If this is negative, then this is negative. If this is negative, then this is negative. And so what we see here is that this gets more and more-- becomes larger and larger numbers in the negative direction. If we keep going, if we're thinking about a number line, further and further and further to the left. So we can say the limit of f of x as x approaches 0 from the negative direction is equal to negative infinity. Well that's interesting. Now let's think about a limit as x approaches either positive or negative infinity. So let's now think about the limit of f of x as x approaches infinity. And one way to set up this table-- we can just say-- do a similar thing. x and f of x-- so if x is 10, then f of x is 1 over 10. If x is-- and I'm just going to go larger and larger numbers-- if x is 1,000, then f of x is 1 over 1,000. If x is 1,000,000, then f of x is going to be 1/1,000,000. So you see as x gets larger and larger and larger in the positive direction, this f of x now gets closer and closer and closer to 0. So we can say the limit of f of x as x approaches infinity is equal to 0. Now let's think about the limit of f of x as x approaches negative infinity. So we're going to take numbers that are more and more and more negative. Well if x is negative 10, this is going to be negative 1/10. If x is negative 1,000, this is going to be negative 1/1,000. If this is negative 1,000,000, this is going to be negative 1/1,000,000. But we still see that we are approaching 0. So here, once again, we are once again approaching 0. So what implications does this have, besides that we've just been able to deal with limits. And once again, I haven't given you a formal definition of this, but it's hopefully giving you an intuition as we take limits to infinity to negative infinity-- actually this is supposed to be negative infinity-- limits to infinity, limits to negative infinity, or when our limited self is infinity, or negative infinity. So one, we're seeing that we can do that. But let's actually try to visualize this when we look at the graph of f of x is equal to 1 over x. So let's do it-- actually maybe I want to be able to keep looking at all of this stuff, so let me set up the graph. right over here. So that's our x-axis. This right over here is our y-axis. And let's graph f of x. So we see that if x is a very small number, if x is 0.1, then y-- y is equal to f of x-- is going to be a very high number. And as the closer and closer we get to 0 from the positive direction, f of x approaches infinity. So it just keeps approaching infinity as we get closer and closer to 0. As x gets closer and closer to 0, the y value just gets higher and higher. Then, as our x value gets larger and larger, our f of x value gets smaller and smaller. So it looks something like that it approaches 0. Similarly, if we approach x from the negative direction right over here, we saw that f of x is approaching negative infinity. So as we get x is closer and closer to 0, our f of x gets more and more and more negative. And then as our x becomes more and more negative, the x itself becomes more and more negative. We see that our function is approaching 0. So the way I've drawn it, we see that there's actually a two asymptotes for the graph of f of x is equal to 1/x. You have a horizontal asymptote at y is equal to 0. When x approaches infinity, f of x gets closer and closer to 0 but never quite touches it. When x approaches negative infinity, f of x is getting closer and closer to 0 from the bottom but it never quite touches it. And we also have a vertical asymptote right over here at x is equal to 0. And we see that because as x approaches 0 from the positive direction, y approaches infinity. And as x approaches 0 from the negative direction, y approaches negative infinity. So the limit here, at x is equal to 0-- so if you were to say, we looked at the limit as x approaches 0 from the positive direction and from the negative direction, but we see that they're approaching two different things. So we definitely have a vertical asymptote at x is equal to 0. But the limit as x approaches 0 of f of x-- this is not defined. Why is that? Well, when we approach 0 from the positive direction we get a different thing than when we approach it from the negative direction.