First we set...
... and imnmediately we obtain...
Since the limit depends form the limit itself doesn't exist...
Kind regards
The limit...
... exists if and only if tends to the same value no matter which is the 'traiectory' [straight line, zig zag, spiral, etc...] with which tends to ... this is the fundamental concept of limit of two [or more...] variables functions...
Kind regards
Thanks allot, i just have another question,
Is the polar coordinate substition the best way to solve limits of 2 variables? or just one of the many? I have tried to use the formal definition of a limit, but I end up confusing myself..
For exmaple, I tried using the formal definition on that function, and dont go anywhere. Another one I was working on;
So using the formal definition;
Then i stated that
But i wasn't sure where to take it form here, or even if this is on the right track...
OMG.. thanks allot. I tried using the polar coordinates, but forgot the [/tex]\rho[/tex] from the denominator. So the condition is, that if is left in the function after substituting in the polar coordinates and simplifying then the limit exists?
The text i have is very light on using polar coordinates, it just has the sentenace (after using polar coordinates for a limit) that 'the behaviour as depends on hence there is no limit'
Sorry for all the questions, i am just trying to get my head around this in a logical manner :P
You can examine a few paths and see if you keep getting the same result.
I like using y=mx, then let x head to zero.
OR try...
(1) y=o and let x tend to zero, the limit along both parts of the x-axis as we head to the origin is 0, since you have 0 over .
And for a second path, just let x=y, which is just calculus one now...
(2) In this case the limit is
SINCE two different paths produce two different limits, the limit does not exist as we approach the origin.