Question : Determine whether the limit exists for the following. If so, find their value.
$\lim_{ ( x , y ) \to ( 0 , 0 )} (1 − x ^2 − y^ 2)/( x^ 2 + y ^2)$
Normally we solve this by either cancelling common terms in numerator and denominator, or by finding the limit along another lines passing through the same common point $(0,0)$. Which gives different limits hence limits won't exist.
This was my approach:
(i)Taking limit along line $y=x$: $\lim_{ ( x , x ) \to ( 0 , 0 )} (1 − 2x ^2 )/( 2x^ 2)$ Substituting $x=0$, we get limit approaches $\infty$.
(ii) Taking limit along line $x=0$: $\lim_{ ( 0 , x ) \to ( 0 , 0 )} (1 − y ^2 )/( y^ 2)$ Substituting $y=0$, we get limit approaches $\infty$.
(iii)Taking limit along line $y=0$: $\lim_{ ( x , 0 ) \to ( 0 , 0 )} (1 − x ^2 )/( x^ 2)$ Substituting $x=0$, we get limit approaches $\infty$.
Hence we observe that the limit exists through any line through the common point $(0,0)$ and that limit tends to $\infty$.
Final conclusion : Limit exist and that limit equals $\infty$.
My doubt is that if we get limit as $\infty$ like this generally, should we rather say that the limit doesn't exist ? (As $\infty$ is not defined).
Is my approach and conclusion correct?