Question

Find the two repeated limits of the function f(x,y)=(y-x/y+x)(1+x^2/1+y^2) at (0,0).Does the simultaneous limit of f exist as (x,y) -(0,0)?Give reasons for your answer

Answer 1

The given function is

$f(x,y)=\frac{(y-x)}{(y+x)} \frac{(1+x^2)}{(1+y^2)}$

The repeated limits of $f(x,y)$ are following:

$\lim x\to0(\lim y\to0 \space f(x,y) )=\lim x\to0(-1)(1+x^2)$

$=-1$

$\lim y\to 0(\lim x\to 0 \space f(x,y))=\lim y\to0 (\frac{1}{1+y^2})$

$=1$

Simultaneous limit of $f(x,y)$ at $(0,0)$ does not exist because repeated limits are not equal.