Question

Can someone help me find the limit of this multivariable?

Answer 1

limit of arctan (-1/(x^2+y^2))

Answer 2

x and y going to 0 I don't know how to fix the denomintator

Answer 3

that's like saying lim x-> -infinity arctan(x), right?

Answer 4

which happens to equal -pi/2.

Answer 5

Wait how did you get that?

Answer 6

it's in terms of (x,y)-->(0,0)

Answer 7

Yes, but it's the same limit, 1/(x^2+y^2) gets large as x and y approach zero.

Answer 8

okay could you help me with lim exp(-1/(x^2+y^2)) x,y goes to 0

Answer 9

the inner bit is the exact same, I just want to know how to properly reduce it

Answer 10

would I do the same and say lim x--> - infinity is exp(x)?

Answer 11

Maybe I was wrong about the arctangent, I'm not sure.

Answer 12

I would first try reducing the inner bit by partial fraction decomp and then reanalyzing the limit.

Answer 13

partial fraction decomp?

Answer 14

can we some how use polar coordinates?

Answer 15

Uhp, nevermind, that's a "+". Perhaps you can divide everything by "x^2"?

Answer 16

? I'm slightly confused you mean divide it over arctan?

Answer 17

No, manipulate the fraction.
arctan( (-1/x^2)/(1 + (y^2/x^2)) )
like we used to do in elementary calculus. This gives us lim arctan( (0)/(1 + (0) ) which then gives us the lim as arctan goes to 0.

However, I'm not sure how valid this is, might want to get more references.

Answer 18

also, the limit of (y^2/x^2) would not be 0, it would be 1 by l'hospital's rule, I do believe.

Answer 19

I think it's -pi/2 using polar coordinates which gives lim arctan (-1/r) r-->0+

Answer 20

in retrospect. I think polarizing the coordinates would be the way to go here. Sorry, I completely forgot about that.

Answer 21

yeah thanks for trying