# Prove the limits exist or DNE

1. Oct 6, 2008

### pantin

1. The problem statement, all variables and given/known data

lim (x,y,z)->(0,0,0) (xy+yz+xz)/(x^2+y^2+z^2)

lim(x,y) - > (1,1) (x-y^4)/(x^3-y^4)

lim(x,y)->(0,0) [1-cos(x^2+y^2)]/(x^2+y^2)^2

one more..

is there a constant c in R s.t. the fn

f(x,y) =( xy+y^3 ) / (x^2+y^2) for (x,y) =/= (0, 0)
f(x,y) = c for (x,y) =(0,0)

is continuous at (0,0)?

2. Oct 6, 2008

### HallsofIvy

Staff Emeritus
What work have you done yourself? For example, if a limit in 2 or 3 variables exist, then the limit must be the same as you approach the point from all directions. Have you tried seeing what happens if you approach along, say, x= y= z or y= x2?

That will only prove that a limit does NOT exist. To show that one does exist or to actually find the limit, try putting it into polar coordinates. That way r alone measures the distance to the origin.

3. Oct 6, 2008

### tiny-tim

Hi pantin!
Hint: this obviously DNE … consider various ratios of x y and z.
Show us what you've tried, and where you're stuck, and then we'll know how to help.