Question

Evaluate the limit or show it does not exist.

Answer 1

\[\lim_{(x,y) \rightarrow (0,0)}\frac{ x^2-y^2 }{ x^2+y^2 }\]

Answer 2

Just show it approaching from different paths?

Answer 3

Try polar coordinates.

Answer 4

Makes it harder.

Answer 5

I get:

\[\lim_{r \rightarrow \theta} \cos^2(\theta)-\sin^2(\theta)\]

Answer 6

r approaches 0 sorry not theta.

Answer 7

Yeah, which is equal to \[
\\cos^2(\theta)-\sin^2(\theta)
\]Now convert it back to Cartesian coordinates.

Answer 8

Ohh... :P .

Answer 9

Sec.

Answer 10

Actually that doesn't really help, nevermind.

Answer 11

I can't though.

Answer 12

Yeah :P

Answer 13

I think evaluating it to \[
\cos^2(\theta)-\sin^2(\theta)
\]Does have some meaning though.

Answer 14

No I got a better idea actually.

Answer 15

Got it!

Answer 16

Try the path y=x and y=0. THe limits dont match so the limit does not exist.

Answer 17

Thanks though :) .