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) to (0,0)?Give reasons for your answer

Answer 1

The given function is ,

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

Now, we have to find the repeated limit of $f(x,y)$

$\lim_x\to0$ ($\lim_y\to0 f(x,y)$ )$=\lim_x\to0$ $($ -1×($1+x^2)$ $)$

$=-1$ .

$\lim_y\to0$ $($ $\lim_x\to0f(x,y)$ $)$ $=\lim_y\to0$ $($ $\frac{1}{(1+y^2)}$ $)$

$=1$ .

No,simultaneous limit of f at (0,0) does not exist.

Because the above repeated limit are not equal.