Vector Calculus

[QuantumJG]

I'm starting a forum for my maths subject Vector Calculus and here is my first question.

Show that the following limit does not exist:



I tried proving it by going along the path

So

Proof:





So the limit doesn't exist since the k dependency implies that the limit value will depend on what parabolic path you take.

But I'm not sure if this is right.  :uglystupid2:

[TrueTears]

Hi there, this is my working:

Let's approach (0,0) from the x axis.



Now let's approach (0,0) from y axis.



Seems good so far...

Now let's approach (0,0) from the straight line y = mx



So all 3 path lead to 0, now let's try parabolas.

The rest follows from your working.

It's correct.


However I'd change your final sentence to this: where C is an arbitrary constant. So we have find a path which leads to a different limit. Thus the limit at (0,0) does not exist.

To be even more rigorous, you can try a proof if you want. That'd be an overkill, but fun

[QuantumJG]

Hi there, this is my working:

Let's approach (0,0) from the x axis.



Now let's approach (0,0) from y axis.



Seems good so far...

Now let's approach (0,0) from the straight line y = mx



So all 3 path lead to 0, now let's try parabolas.

The rest follows from your working.

It's correct.

Thanks TT!

In the proof I probably should start with linear equations and then move to higher order polynomials. Doing limits for functions of several variables is a bit more trickier because you aren't constrained to approaching the limit from the left or right, but you can approach it in more ways than you can imagine.

[TrueTears]

Yup, also if y=kx^2 doesnt work try x = ky^2 and if that also leads to a same limit, you should think that the limit might actually exist and use an e-d proof to show it does.

[Ahmad]

Just as an aside in the way of counter-examples, there are limits that don't exist but for which if you approach along any polynomial path you get the same value!

[Cthulhu]

Just as an aside in the way of counter-examples, there are limits that don't exist but for which if you approach along any polynomial path you get the same value!

I ran into this last semester. Confused the hell out of me.

[QuantumJG]

Just as an aside in the way of counter-examples, there are limits that don't exist but for which if you approach along any polynomial path you get the same value!

Thanks I'll keep this in mind.

[Ilovemathsmeth]

WOW I'd love to do this subject. I don't think Actuarial requires it though

[tram]

ran into this question (or a very similar question) in umep....i don't get limits.... like, how do you know how you know which path you should approach the limit from to prove it dosen't exist? and if it's just a proof by contradiction, if the limit DOES exist, how do you prove it exists cos it's not like you can go though every single possibilty it and prove they ALL work.....

is there a thread explaining limits somewhere? soz to be all nooby >.<

WOW I'd love to do this subject. I don't think Actuarial requires it though

Vector caculus is the subject they recommend you do in first semester for acturial if you've done umep. Thus it would be highly likely that some acturial studes people have done this subject.

[kamil9876]

there are some techniques which account for every possible path.

For example if I want the limit as (x,y) approaches 0 of:


I bound it by:



and apply sandwhich theorem. (also to be rigorous you should mention that my inequality doesn't make sense when x=0, but when x=0 the function value is 0 anyway).

Read a precise definition of limits to remove any confusion.

[TrueTears]

ran into this question (or a very similar question) in umep....i don't get limits.... like, how do you know how you know which path you should approach the limit from to prove it dosen't exist? and if it's just a proof by contradiction, if the limit DOES exist, how do you prove it exists cos it's not like you can go though every single possibilty it and prove they ALL work.....

is there a thread explaining limits somewhere? soz to be all nooby >.<

WOW I'd love to do this subject. I don't think Actuarial requires it though

Vector caculus is the subject they recommend you do in first semester for acturial if you've done umep. Thus it would be highly likely that some acturial studes people have done this subject.

what i do is this, check your normal paths, x/y axis, linear lines, parabolas, if those all yield same limit then try an proof to see if the limit is actually what you conjecture. That is as rigorous as it gets.

[QuantumJG]

ran into this question (or a very similar question) in umep....i don't get limits.... like, how do you know how you know which path you should approach the limit from to prove it dosen't exist? and if it's just a proof by contradiction, if the limit DOES exist, how do you prove it exists cos it's not like you can go though every single possibilty it and prove they ALL work.....

is there a thread explaining limits somewhere? soz to be all nooby >.<

WOW I'd love to do this subject. I don't think Actuarial requires it though

Vector caculus is the subject they recommend you do in first semester for acturial if you've done umep. Thus it would be highly likely that some acturial studes people have done this subject.

In Vector Calculus we use the sandwhich theorem. Basically you find a function that is always less than your function and one that's greater than your function and you squeeze your function inbetween them and use it to prove the limit exists.

ε-δ proofs I just hate and apparently in vector calculus you aren't required to use them. Although I should probably refresh my memory on them.

[tram]

*brain explodes*

ok i'm going and doing some reading on that e-d thingy proof and the sandwich theorem THEN comming back to annoy you guys

[Mao]

Simple answer to your 'what is a limit', it is a value you get as you approach a particular coordinate in a [possibly multivariable] function. Here, the keyword is 'approach'. You may want to know which way we approach it, and the answer to that is EVERY POSSIBLE DIRECTION. For a limit to exist, you must approach the same value from every direction.

For a univariable function (1D), you can only approach a coordinate from left and right. If you can show they go to the same value, then you are done.

For a multivariable function (2D or higher), you can approach a coordinate from an infinite number of paths, thus you can't use the above method. What we end up employing is the epsilon-delta proof, which incorporates the infinite number of paths. However, it is tedious, algebraically intensive and often very confusing, so we don't always want to do an epsilon-delta proof straight away. Instead, we test a few simple paths, which is much simpler than the e-d proof to try to find a counter-case (thus proving a limit does not exist), and if we cannot find a simple counter-case, we do a e-d proof.

[QuantumJG]

Using the matrix chain rule evaluate the derivative of:

f(g[f(x,y)]) at (0,1) where f(x,y) = (x², 2y, x - y) & g(x,y,z) = (x + z, y²) 

I really need help since I'm finding this area hard.