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limit of a function of two variables watch

1. hi, i don't even understand how he started with this question if anyone could just help with a few chunks of it it'd be a massive help! thank you
2. Switched from cartesian to polar co-ordinates and used the expansion of "cos p" and "sin p", where "p" is just whatever's within the function.
3. Are you familiar with the Trig Taylor Expansions and with "Big-Oh" notation? Give it a wiki if not.

The first bit of paper could use on extra step in the middle. Write down the Taylor expansions for both sine and cosine (making sure you are considering what the argument is, so set xy=z then do it around z etc.) Then you see you have some higher order terms (H.O.T) which get smaller far quicker than the initial terms. The big O basically groups all of these terms together, saying that "The Higher order terms will act like me in the limit". If you're careful it means, when considering the limit, you can chuck the Big-Ohs away

Can you understand the rest of it from here?
4. (Original post by KAISER_MOLE)
Are you familiar with the Trig Taylor Expansions and with "Big-Oh" notation? Give it a wiki if not.

The first bit of paper could use on extra step in the middle. Write down the Taylor expansions for both sine and cosine (making sure you are considering what the argument is, so set xy=z then do it around z etc.) Then you see you have some higher order terms (H.O.T) which get smaller far quicker than the initial terms. The big O basically groups all of these terms together, saying that "The Higher order terms will act like me in the limit". If you're careful it means, when considering the limit, you can chuck the Big-Ohs away

Can you understand the rest of it from here?
ok bare with me! i've read about the taylor expansion, and tried it for cos(xy)-1. i came out with

(xy)^2/2! + (xy)^4/4!

the only difference with his answer was that the first (xy) had a minus in front, which i don't know how he got. and the second part is 0((xy)^4) so i'm guessing this is where the "big O" comes in. i don't really understand the concept of this, why can't i just leave it as what i got? (sorry i know i'm really annoying!) thanks for the help!!
5. I think the minus sign is supposed to be there (check your series again, should be cos(xy)=1-0.5(xy)^2+... or something like that, so cos(xy)-1 leaves the minus sign there)

The Big O is there as you still have to include all the terms that come after (xy)^4 (there are infinitely many of them!). So as you approach (0,0), the behaviour of the higher order terms is that 'It converges to 0 at least as fast as (xy)^4'

Why can you group all the terms together like this? Is it true to say, considering the whole lot of them, they act like (xy)^4? Well, each of the terms has a factor of (xy)^4 in them, so take that out and you're left with (xy)^4(1-a(xy)^2+b(xy)^4-...) where a and b are just the coefficients. This term in the brackets is certainly finite for |x| and |y| both less than a half say (since cos(xy)-1+0.5(xy)^2 is bounded above and below). So you can group all these terms together

Why you do this should become apparent if you follow the rest of the calculation.

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