Rigorous Multivariable Limit Definition Problem

[dumbdumNotSmart]

Homework Statement

Hey I'm trying to prove the rigorous definition of limit for the following function:
Lim (x,y) approaches (1,1) of f(x,y)=(y*(x-1)^(4/3))/((x/1)^2+abs(x)*y^2)

Homework Equations

abs(x^2)<abs(x^2 +y^2)

The Attempt at a Solution

I know the rigorous definition of limit. I tried to constraint the denominator by eliminating one of the terms since both are greater than zero, however, I was left with what seemed like terms that could not be constrained. I never really done a rigorous definition of limit that's not centered on (0,0). I would appreciate some help.

 

[Lucas SV]

If you feel more comfortable with limits centred at ##(0,0)##, you can deform the problem (in order to get an idea for ##\delta##), by doing a shift ##(x,y)\rightarrow (x-1,y-1)##. I'm assuming proofs are required to be directly from the ##\epsilon##-##\delta## definition?

 

[Mark44]

Homework Statement

Hey I'm trying to prove the rigorous definition of limit for the following function:
Lim (x,y) approaches (1,1) of f(x,y)=(y*(x-1)^(4/3))/((x/1)^2+abs(x)*y^2)

Typo above? Should the (x/1)^2 be (x - 1)^2?

Homework Equations

abs(x^2)<abs(x^2 +y^2)

The Attempt at a Solution

I know the rigorous definition of limit. I tried to constraint the denominator by eliminating one of the terms since both are greater than zero, however, I was left with what seemed like terms that could not be constrained. I never really done a rigorous definition of limit that's not centered on (0,0). I would appreciate some help.

 

[dumbdumNotSmart]

Typo above? Should the (x/1)^2 be (x - 1)^2?

Yes, that is in fact a typo.

 

[dumbdumNotSmart]

If you feel more comfortable with limits centred at ##(0,0)##, you can deform the problem (in order to get an idea for ##\delta##), by doing a shift ##(x,y)\rightarrow (x-1,y-1)##. I'm assuming proofs are required to be directly from the ##\epsilon##-##\delta## definition?

That's right, however I may constraint the whole thing backwards, that is, work from ε and thereafter find δ in the form of ||(x,y)-(v,w)|| inside the ε I am constraining.(which in the definition of the limit is |f(x,y)-L|<ε L being the Limit I am trying to prove)... I'm not sure how much sense that made, let me know if you need any more clarification. The course is not in english, as you might have noticed.