find lim (x,y) - (0,0) of (x+y)(1/x-1/y)

Question

find lim (x,y) -> (0,0) of (x+y)(1/x-1/y)

anonymous · Answer

$$\lim_{x,y \rightarrow (0,0)} (x+y)(\frac{ 1 }{x } + \frac{ 1 }{ y})$$

amistre64 · Answer

x+y
1/x - 1/y


 x^2y + xy^2
       y - x

might not be easier, but at least gets rid of the fractions

anonymous · Answer

so, i simplied to: y^2-x^2/(xy), I then used polar coordinates and got to r/cos(theta)sin(theta)

amistre64 · Answer

the limit of a multivariable only exists if its the same from ALL directions ... which can be an utter pain to assess

anonymous · Answer

not sure what you did above?

amistre64 · Answer

mulitplied top and bottom by xy

amistre64 · Answer

and your post has -, but the latex has +

amistre64 · Answer

lol, i thought i saw a division bar in the ascii :/

anonymous · Answer

my bad, its minus

amistre64 · Answer

my idea is to test the limit in key directions,  like y=x and such

anonymous · Answer

ye, thats a good idea.I was trying to pick a direction to contradict eachother.

amistre64 · Answer

$$\lim_{x,y \rightarrow (0,0)} (x+x)(\frac{ 1 }{x } + \frac{ 1 }{ x})=1$$

anonymous · Answer

its a minus, remember

anonymous · Answer

lim (x,y) -> (0,0) of (x+y)(1/x-1/y)

amistre64 · Answer

ugh, too early to remember ...

anonymous · Answer

lol :)

anonymous · Answer

maybe y = 2x. in (x^2y + xy^2)(y - x)