in this video we're going to explore the idea behind limits they are it's actually a very easy topic um there's a lot of in-depth maths in the background as to what's really happening but i've seen the way that they test the exams for grade 12 and the way they explain it in class there's an easy way to do this the more in-depth stuff they only introduce typically in university level so all you do is you plug the limit value into this expression and if you don't end up with any errors on your calculator then everything is fine and so what you can do is you can just try plugging it in first you could just say equals to zero squared plus three times zero plus four and if you do that on the calculator the answer is four and that's it that's how you do limits obviously there are going to be many instances where that doesn't work but always try that out first so in this example we should always try plug in the value straight away so let's do that so we'll say like that over there but then when you press equals on your calculator you're going to get an error now the reason for that is that at the bottom over here if you're plugging -1 into the place of x well you ending up with a denominator of zero and that is the main area that limits will catch you is when the denominator becomes zero but then if you get a question like this in the test it has been designed in such a way that the denominator can be cancelled so let's see how that would so now we're not going to plug minus one in yet so we're going to carry on with the questions so now here's where teachers get very or the department is very strict about this you have to keep saying equals to limit x goes to -1 and you have to keep writing that until you actually use the minus one and sorry if this doesn't if this isn't clear what i'm doing here it says x and then there's an arrow and the arrow is pointing to minus one so it says x is going to minus one okay so now what we're going to do is we're gonna have to do a few things so what could we possibly do at the top well well done if you realize that we can factorize and so that's a nice easy trinomial they'll always design them as fairly easy if they need the denominator to cancel so that's just going to be x minus 4 x plus 1 over x plus 1 and so that cancels and so then we can just write equals to lum x goes to -1 of x minus 4. now we can put the minus 1 into the equation so then we can just do this now you don't have to say lim anymore because we're going to use the minus 1 like that and the answer is minus 5. see so it's easy if you get an error on your calculator then you just have to do an extra step and then just plug the value in okay so here we can try plug everything in but we can clearly see that there's a zero at the bottom and so if you just plug x as sorry there's an x at the bottom and if you plug zero into that you're going to end up with a zero and you kind of have a zero in the denominator so then all that you do is you factorize then we're going to have to say equals to lum x goes to zero then you take out x as a common factor at the top and then you're left with x plus 3 the x's then cancel and so we can say equals to lum x goes to 0 of x plus 3. and now we can just plug the value in and so that's going to give us 0 plus so that's in bracket plus 3. and 0 plus 3 is just 3. so the answer for that one is 3. so in summary plug the value in if it doesn't work factorize or do something that will help it to somehow yeah they'll only be one or two things that you can possibly do you simplify it like that and then you plug the value in and if you don't get an error on your calculator then it's all fine here's another one so we'll start by just plugging the value in and if you plug it in at the if you plug one in at the top you end up with one minus one but then at the bottom you end up with one plus one now this is okay because we end up with a zero at the top but a two at the bottom that's fine 0 divided by 2 is just 0. what you don't want is you don't want a 0 at the bottom that's where the error comes in on the calculator and that's when you're going to have to do something else and here's one last one so if we plugged two into this expression well at the bottom you would have two minus two which is zero now that's a problem and your calculator would have given you an error so then we have to do a few things so now because we're not using the two yet we have to keep writing this limb x goes to 2 that's very important then at the top we have a difference of squares it can become x minus 2 x plus 2 over x minus 2. those can cancel and so we're still not using the 2 yet and that's going to say x plus 2 now we're going to use the 2 so we don't have to write lim anymore so we just say 2 plus 2 and that's going to give us 4.