Introduction to Limits

Added: 2026-05-05

[00:00:00]Back to the channel. So, today we're going to be uh moving away from what we've been learning from the last few videos and now we're going to be switching complete topics and now we're going to limits. So, um let me kind of explain what like limits are. So, limits are So, limits are essentially if you like look at a certain point right on the x-axis, right? Um let's say like um or okay, let's take like the point like x equals let's say like four, right? So, on on a graph, right? This would be like here, right? So, it um so, limits would be how where what the graph is doing as um uh the given function is approaching this point uh from uh different sides, right? So, you know, that's kind of what it's uh that's kind of what limits are, right?

[00:00:49]So, uh here if x equals four, right?

[00:00:51]Here from this graph like visually just like as a guess, right? You'd say that um uh not not as a as a guess, but like from what you can just see visually, right? Without any data, um uh we can see that from the left side, right? So, if we but if we when you say from the left side, right? We're saying from left to right, what's happening as we approach this point and from right to left, what's happening when you approach this point, right? And so, if you look from left to right, it looks like it's going up to infinity and when you go from right to left, it also looks like it's going to infinity, right? If we change this graph a little bit, right?

[00:01:20]If we maybe made this uh something like this, right? Um then this graph would be uh going again from left to right to infinity, but then from right to left, it'd be going to negative infinity.

[00:01:33]Uh so, you know, that that's how you think about limits, right? So, here uh we have um data points, right? So, we have like x and y, right? So, if x is just y. So, So, here we're being asked what the limit of f of x is as x approaches two from the left and that's what that minus sign up there means. That means from the left. Positive means from the right.

[00:01:53]That's pretty easy to think about, right? Cuz left on the uh if you go like more and more left on uh coordinate plane, it's going to get negative, right? The more and more negative. As you go right, more and more positive. So, that's how you can kind of think about it, right? And then here this has no sign because that's just saying what happens at two. So, like um so, how we just discussed the last example, right? Where it's going to four. So, uh it was either going to infinity or negative infinity from left and right. So, it was at different places, right? So, that's how um that's what it's asking, right? What the exact point is. So, I'll come to that, but um for now um we can just start on this problem. So, So, what are we looking at here? Well, as x values are increasing, right?

[00:02:34]Getting closer and closer, right?

[00:02:36]Infinitely closer to two from the left, right? So, from the left, we're saying like uh less than two, right? So, coming from less than two, closer and closer to two, right? So, 1.9, 1.9, right? So, getting closer to two, right? And what can you see is happening? So, this is um subsiding to 0.25, right? So, you have 0.2564, right? A greater than two is 0.2 25. So, that you can round this up to 0.26 here, right? Here you can round that up round that up to 0.26, right? Um here it'll be 0.25 rounded. Here it's even closer 0.25. So, you can see that it's coming to 0.25 and you can kind of say that as what by the time it approaches two, it will be 0.25 cuz that's what our data set is showing.

[00:03:19]Okay. And now we have to look at what's happening from the right, right? So, then we kind of look at the opposite way, right? Like a greater like the farthest most point from two, what's happening as you get closer to two, right? That's going to be like from 2.1 to 2.01 to 2.0001, right?

[00:03:34]So, it's getting closer to two now, right? And what can you see here? It's like a similar pattern, right? So, 2439, right? Pretty far away from 0.24, right?

[00:03:41]If you just look at it, right? And then you got 2494.

[00:03:45]Um Oh, sorry. Wait on. Sorry. Uh if you like here, you can get like 2414, right?

[00:03:52]Right? So, getting closer and closer, right? As you get closer to um 2.001, 0001, right? So, you're getting more and more specific and at 2.0001, the closest to two on this um on this data on this data set, you find that if you round that up, it's it's basically 0.25, right?

[00:04:12]Here also.

[00:04:13]So, now uh we come to where it's equal to two. Well, um so, there's two cases here. So, if the value that you find from the left and right is exactly the same, then it's just going to be that value, right? But um if this was maybe some other value, right? So, how we discussed the uh it going like the point where it's like uh x is uh going to four, right? Where we drew that example in the beginning. As x goes to four, right? Like this point here, right? We drew like it's increasing to infinity, but then also it's reducing negative infinity. Since from the right and from the left, it's going to different points, uh we're going to write undefined if that if that happens. So, um but since here it's the same, it's going to be 0.25. And that's it. Thank you.