Answer to Question #107181 in Calculus for Nikesh gautam pandit ji

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

$\lim_{y\rightarrow0} \lim_{x\rightarrow0} \dfrac {y-x} {y+x}\cdot \dfrac {1+x^2}{1+y^2}= \lim_{y\rightarrow0} 1\cdot \dfrac {1}{1+y^2}=1$

$\lim_{x\rightarrow0} \lim_{y\rightarrow0} \dfrac {y-x} {y+x}\cdot \dfrac {1+x^2}{1+y^2}= \lim_{x\rightarrow0} -1\cdot \dfrac {1+x^2}{1}=-1$

The repeated limits don`t coincide so the simultaneous limit doesn`t exist