Hello,
Have you tried converting all rectangular coordinates into polar coordinates ??
find the limits by referring details when you solve them when (x,y)goes to (0,0) :
1_ lim (x^2 + y^2 ) sin ( 1\ (x ^2 + y^2))(i'm trying but i don't know )
2- lim e ^ -1 over (sqr root for x^2 + y^2)
3- lim { e ^ -1 over (sqr root for x^2 + y^2) } over (sqr root for x^2 + y^2)
4- lim 1- ( x^2 + y^2) \ ( x^2 + y^2)
5- lim y ln ( x^2 + y^2)
when i solve the limit #4 first i substitute (0,0) the answer 1\0 i know that the limit doesn't exist but i don't know how i can show that ???
in #5 when i translate xy-coordinate to r,theta i can't complete
I'll help you on the first one because your notation is a little off on the rest:
(x^2+y^2) sin(1/(x^2+y^2))
Let x^2+y^2=r^2 through parametric transformation:
(r^2) sin(1/(r^2))
Now if we know that r=0 (because x^2+y^2=0), then we know we have:
0* sin(1/(r^2))
Regardless of what the sin value produces, you know it is within the limits of the sine function solution (-1 through +1), and because zero times any constant, even in this case zero, should the sine value equal zero, would still be zero^2=0.
Can you get it from here?