How to solve: Evaluate lim (x,y)→(0,0) sin x. cos(1/y)

[Ind]

I am not sure, how to prove the limits does not exist.
I did the below steps:
I calculated the limits along the x axis:
Lim x->0 f(x,0) = lim x->0 sinx.cos(1/0) = sinx.[infinity] = sin(0).infinity = 0.
Similarly along y axis.
lim y->0 (sin (0).cos(1/y)) = 0.

Could you please help me understand this problem. - Thanks!

 

[tkhunny]

I am not sure, how to prove the limits does not exist.
I did the below steps:
I calculated the limits along the x axis:
Lim x->0 f(x,0) = lim x->0 sinx.cos(1/0) = sinx.[infinity] = sin(0).infinity = 0.
Similarly along y axis.
lim y->0 (sin (0).cos(1/y)) = 0.

Could you please help me understand this problem. - Thanks!

#1 "Approach" does NOT mean "at". Do NOT substitute until AFTER you know it's continuous.
#2 Are you SURE "0 * infinity" is zero? Please lookup "Indeterminate Form".
#3 You cannot approach your goal along a path that is ENTIRELY without the Domain. Please rethink "along the x-axis".
#4 Prove ANY TWO valid pathways result in different conclusions. That's one way to prove it doesn't exist.