Rigorous Multivariable Limit Definition Problem

  • #1

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.
 

Answers and Replies

  • #2
140
50
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?
 
  • #3
35,009
6,759

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?
dumbdumNotSmart said:

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.
 
  • #5
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.
 

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