Limits of multivariable functions: lim_(x,y->0,0) (x^2+y^2)/(10x^2+2xy+5y^2), lim_(x,y->0,0) ye^(-1/r) w/ r=sqrt(x^2+y^2)

[Nox]

I am supposed to check if the limits exist for these 2 functions and if they do I am asked to give the value it approaches

I converted a) into polar coordinates and found the limit to be 1/10 but in the solution manual it says the limit does not exist and in the manual a different method is used, they use paths

In b) I used polar coordinates and the limit exists and its zero and the solution manual has the same method

Why is it okay to convert to polar coordinates in b) and get the answer that way but not a) ?

 

[Nox]

My solution to a)

 

[blamocur]

Why is it okay to convert to polar coordinates in b) and get the answer that way but not a) ?

Just converting to polar coordinates does not prevent you from making mistakes.

 

[blamocur]

My solution to a)

Why are you assuming that [imath]\theta=0[/imath] ?

 

[Nox]

I see the problem, I assumed θ = 0 when infact θ can equal anything as long as r=0. So the fraction I got will approach different limits depending on θ and therefore the limit does not exist. Is my reasoning correct?

 

[blamocur]

, I assumed θ = 0 when infact θ can equal anything as long as r=0.

Sounds mostly right to me, but I would phrase it differently: [imath]\theta[/imath] can be arbitrary as long as [imath]r\rightarrow 0[/imath] to make [imath]x,y\rightarrow 0[/imath]. I.e., fixing [imath]\theta[/imath] produces different limits.