If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains ***.kastatic.org** and ***.kasandbox.org** are unblocked.

Main content

Current time:0:00Total duration:7:41

let's say that f of X is equal to one over X and we want to think about what the limit of f of X is as X approaches zero 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 what f of X is going to be and I'm going to approach X I'm going to approach zero from the positive direction so let's say we'll try zero point one then we're going to try 0.01 then we can try 0.001 then we could try 0.0001 so notice each of these numbers they're all larger than than zero and they're approaching zero from the positive direction we're getting closer and closer and closer to zero so when X is zero point one f of X is just going to be one over this this is 1/10 so one over that is just going to be ten one over point one over zero point two zero one is going to be 100 one over zero point zero zero one is going to be 1,000 one over zero point zero zero zero zero zero zero one is going to be ten thousand ten thousand so you see as X gets closer and closer to zero 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 zero 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 zero point one two three four five six seven digits behind the decimal place then one over that's going to be one with one two three four five six seven zeroes did I do that right here I had four places behind the decimal four zero so you have one two three four five six seven and here I have seven zero so you see as we get closer and closer to zero 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 the limit as X approaches 0 from the negative direction of that of X or the limit of f of X is 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 zero point zero point one this is going to be negative ten if this is negative then this is negative 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 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 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 the limit of f of X as X approaches infinity and one way to set up this table we can just say we can just say do a similar thing X and X and f of X so if f 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 just 1,000 then f of X is 1 over 1000 if X is 1 million 1 million then f of X is going to be one millionth one millionth 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 zero so we can say the limit of f of X is X approaches infinity is equal to zero now let's think about the limit the limit of f of X as X approaches negative infinity so we're going to take look 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 one thousand this is going to be negative one over a thousand and this is negative 1 million that's going to be negative one millionth but we still see that we are approaching zero so here once again we are once again approaching zero 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 opposed to be negative infinity limits to infinity limits to negative infinity or when our limit itself is infinity or negative infinity so 1 we're seeing that we can do that but let's actually try to visualize this and when we look at the graph of f of X is equal to 1 over X so let's do it maybe I want to be able to keep looking at all of this stuff so let me set up the graph let me set up the graph right over here 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 as X gets if X is very small number if like X is point 1 then Y Y is equal to f of X is going to be a very high number and as a closer and closer we get to 0 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 Y our f of X value gets smaller and smaller so it gets smaller and smaller so it looks something like that it approaches 0 similarly if we approach if we approach X from the negative direction right over here we saw that f of X is approaching negative infinity approaches 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 becomes quote becomes more and more negative the X itself becomes more and more negative we see that our function is approaching 0 its approaching 0 so the way I've drawn it we see that there's actually two asymptotes for the graph of f of X is equal to 1 over X you have a horizontal asymptote at Y is equal to 0 you have a horizontal asymptote at Y is equal to 0 when we when X approaches infinity it gets close f of X gets closer and closer to 0 but never quite touches it when X when X approaches negative infinity f of X is approaching is getting closer and closer to zero from the bottom but it never quite status' it and we also have a vertical asymptote we have a vertical asymptote right over here at X is equal to zero and we see that because as X approaches zero from the positive direction Y approaches infinity and as X approaches 0 from the negative direction Y approaches negative infinity so in the limit here at X is equal to zero so if you were to say we looked at the limit as X approaches zero 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 zero but the limit the limit as X approaches zero of f of X this is not defined not defined why is that well when we approach zero from the positive direction we get a different thing than when we approach it from the negative direction