Can a constant c make the function continuous at a point?

[pantin]

Homework Statement

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)?

 

[HallsofIvy]

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.

 

[tiny-tim]

Hi pantin!

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

Hint: this obviously DNE … consider various ratios of x y and z.

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

Show us what you've tried, and where you're stuck, and then we'll know how to help.