Hello

 

I have the limit

 

lim (x^9 * y) / (x^6 + y^2)^2

 

(x,y)---> (0,0)

 

 

when I use polar the final result is

 

limit =

 

lim (r^6 cos^9 (theta) sin (theta) ) / (r^4 cos^6 (theta) + sin^2 (theta))

 

r--->0

 

and substituting r = 0 , it will give zero

 

* I tried it on wolfram alpha and it gave zero

 

http://www.wolframalpha.com/input/?i=limit+%28x^9+*y%29%2F%28x^6+%2By^2%29^2++as++%28x%2Cy%29+--%3E+%280%2C0%29

 

but when I use cartezean and try the path y = mx^3

 

the result turns to be

m/(1+m^2)^2

 

which depends on m

 

so what is it ?!

Without having checked your calculations: If you have a scenario where the limit depends on the path (which indeed can happen) then no unique path-independent limit -or short: no limit- exists.

I may think that the limit

 

(r^6 cos^9 (theta) sin (theta) ) / (r^4 cos^6 (theta) + sin^2 (theta))

 

has a problem of the sin^2 theta downwards , as it may become zero and the limit won't be zero / positive value any more

 

but then it is 0/0 , so what to do next to know whether this (x/x) will turn eventually to be zero in the limit or not ?!

 

 

And question two , is actually the question in the above comment : why does wolfram alpha say the value of the limit is zero despite the result of m/(1+m^2)^2