I get the jist of it, as a value of x (or any other variable) approaches some number, the function's value will get closer to some other value.
Or is this not correct?
I'm asking as I just want to broaden my knowledge of maths
(Also, how do you USE limits in problems?)
In a nutshell, yeah (though does infinity count as a number? ) Limits are used in things like graph sketching, and they can be used as part of a problem, i.e. evaluating the limit can be the problem. Limits are used when you can't evaluate a problem at an exact point, because problems arise. e,g. x→1limx−1x2
In a nutshell, yeah (though does infinity count as a number? ) Limits are used in things like graph sketching, and they can be used as part of a problem, i.e. evaluating the limit can be the problem. Limits are used when you can't evaluate a problem at an exact point, because problems arise. e,g. x→1limx−1x2
Ah, so that equation there, x can never be 1 because (1-1 = 0, which means you'd have to divide by 0)... but how do you find limits? (or is that hard?)
Ah, so that equation there, x can never be 1 because (1-1 = 0, which means you'd have to divide by 0)... but how do you find limits? (or is that hard?)
x→clim
c can be any number. So there may be easy limits:
x→2limx−120 would simply yield 20 . If you are wondering about finding certain values of c, look for things that would, outside the world of limits, cause headaches
Ah, so that equation there, x can never be 1 because (1-1 = 0, which means you'd have to divide by 0)... but how do you find limits? (or is that hard?)
Well, the example I gave is fairly straightforward, it's just infinity or -infinity depending on the direction of approach to 1. x→1limx−1x2−1 is a different story, a bit of manipulation can yield a limit.
For more complicated limits there's L'Hopital's rule for indeterminate quotients helps to evaluate some. It states that: x→climg(x)f(x)=x→climg′(x)f′(x) Where c is just a constant. The limit is also used in differentiation from first principles.
Well, the example I gave is fairly straightforward, it's just infinity or -infinity depending on the direction of approach to 1. x→1limx−1x2−1 is a different story, a bit of manipulation can yield a limit.
For more complicated limits there's L'Hopital's rule for indeterminate quotients helps to evaluate some. It states that: x→climg(x)f(x)=x→climg′(x)f′(x) Where c is just a constant. The limit is also used in differentiation from first principles.
Woah, you lost me with the second paragraph
So in short, is a limit a value that a function approaches as one of the variables in that function approaches some value.
x→2limx−120 would simply yield 20 . If you are wondering about finding certain values of c, look for things that would, outside the world of limits, cause headaches
Ok, so in your second example, as x approaches 2, 20/x-1 will approach 20?
Ok, so in your second example, as x approaches 2, 20/x-1 will approach 20?
Yes. It's simply a way of writing that the variable will never reach a certain constant (in this case 2), and hence your "answer" will be near 20, which for all reasons and purposes is identical to 20 since it's a limit.
They don't even teach basic differentiation at GCSE, I doubt anyone would understand the proof!
The concept of differentiation and why it works is easy to understand. It's probably on the same difficulty as being able to notice that if you increase the number of sides of a polygon to infinity, your shape starts to look like a circle.
The reason we teach concepts of calculus at a higher level than basic arithmetic is because of the computation.
The concept of differentiation and why it works is easy to understand. It's probably on the same difficulty as being able to notice that if you increase the number of sides of a polygon to infinity, your shape starts to look like a circle.
The reason we teach concepts of calculus at a higher level than basic arithmetic is because of the computation.
I guess, but what about these "Further maths iGCSEs", why instead of limiting teaching slightly higher math to a select few, we can just make the basic concept part of the normal GCSE course,
I guess, but what about these "Further maths iGCSEs", why instead of limiting teaching slightly higher math to a select few, we can just make the basic concept part of the normal GCSE course,
TBH you can start teaching concepts of calculus at primary school, at least without a lot of computation.
Although I'd love to see calculus taught to all GCSE students, I doubt that is going to happen any time soon.
We probably won't start teaching calculus at primary schools before 3000... but basic concepts of calculus can easily be understood at this age. Young child already know that if you zoom really close into a curved line it starts to look straight (Differentiation).
I mean there is basic differentiation and integration in additional maths (FSMQ), that won't be too hard for higher-tier GCSE students at least. But if you look at the standards of GCSE maths papers these days, it's pretty shocking. If we're going to teach any serious calculus at GCSE, then we're going to have to reform primary school education too.
Also as time passes other topics might become more significant. For example a maths curriculum based on problem solving, and not doing tons of arithmetic, might become popular.