I have some problems where I need to determine if the limits exist using the limit definition or limit laws. For the following 2 I think i can use the laws but would like to make sure:

Make y approach zero along the line and then it follows at once that the limit doesn't exist. Tonio
I used the product rule splitting into 2 ie

and

This would give a limit of 0 times infinity = 0 respectively right?

Then the 2nd problem I thought I could handle the same way ie:

I thought i could solve it the same way, ie use the product rule with 2 equations f(x,y)=y and g(x,y) = 1/x giving

I still need to do the proofs now but would like to know if i''m on the right track in terms of at least getting the limits right?

many thanks!