Piece-wise continuity problem...

Hinzugefügt: 2026-05-08

[00:00:00]So, hello guys. So, let's quickly talk about um piecewise function and in fact, continuity.

[00:00:09]Continuity to be precise.

[00:00:11]Now, find the value of K such that f of x is continuous at x equals to 2. Now, before we talk about the continuity aspect, let's check what piecewise is really talking about.

[00:00:22]Now, for instance, if you are dealing with kx squared, you must only be using this condition because it is this condition that is recommended for kx squared.

[00:00:35]You are dealing with 2x + k, this will always be the condition.

[00:00:39]If they ask you when x is tending to 2 from the right, from the right. So, you are using x greater than 2. Because this one that is getting close to 2 from the right.

[00:00:49][clears throat] Something that's probably greater than 2, whether by little or by whatever.

[00:00:56]So, from the right, this all you are using and this the function you are using. If you are from the left or equal to 2, you use this. Use the condition.

[00:01:06]So now, let's talk about continuity.

[00:01:08]Continuity means the limit when a function is continuous, then it means that the left side limit is equal to the right side limit. That is limit as x tends to 2 from the left of f of x equals to limit as x tends to 2 from the right of f of x and they are equal to what? f of 2.

[00:01:34]So, from the left, if it's equal to this, then the limit is continuous. But if it's not equal, the limit is not continuous. So, let's see. We are asked to find the value of K such that f of x is continuous.

[00:01:50]So now, from the left, from the right.

[00:01:56]Now, if I'm looking for the the what this function is approaching from the left.

[00:02:03]If you are looking from the left, rather. If you are looking from the left, that means you are using this function. Because this one is greater than this from the right. This one is the less than equal to. So, this the condition. Even though there's an equal to, but this the only condition that satisfy from the left. Getting close to 2 from the left because it's less than 2. Like 1.9, 1.9999.

[00:02:23]From the left side.

[00:02:24]So, if you are checking from the left, limit as x tends to 2 from the left, we are going to use what? kx squared.

[00:02:35]Because that's the one as I tend to the condition from the left. Which is what?

[00:02:41]K to the square. And that's what?

[00:02:44]4k.

[00:02:46]Right?

[00:02:47]So, we get 4k from the left. Now, from the right, limit as x tends to 2 from the right.

[00:02:54]So, if this greater than 2, this the condition for the right.

[00:02:57]Like 2.01, 2.001. So, we are using this condition.

[00:03:01]2x + k equal to what? 2 x is 2 + k. And what do we have?

[00:03:11]We have 4 + what?

[00:03:13]4 + k.

[00:03:15]Are we together? So, from the left, we have 4k. From the right, we have 4 + k.

[00:03:23]So, are they equal? That's not even the problem.

[00:03:26]We are asked to find the value of K at which they will be equal. Normally, in my own sense, 4k is not equal to 4 + k.

[00:03:34]Isn't it? But it can be equal. It depends on the value of K.

[00:03:38]You get what I'm saying? It depends on the value of K. So, we are asked to find the value of K such that from the left will be equal to from the right.

[00:03:47]Such that f of x is continuous at this.

[00:03:51]So now, 4k, so you are going to equate the two equals to 4 + k.

[00:03:57]Bring this K here. We have 4k minus k equals to 4.

[00:04:01]We have 3k equals to 4. Then k equals to what? 4 over 3.

[00:04:08]For these two to be equal, that means K must be equal to what? 4 over 3.

[00:04:14]Which means if you have kx squared like 4 over 3 x squared and you have 2x + 4 over 3, from the left, which is this, and from the right, which is this, they will be equal. They will be continuous.

[00:04:31]So, that's that for that.

[00:04:33]So, see you in the next video.

[00:04:35]Thanks for watching.