<s>[INST] Proof that $lim_{x->0,y->0}{\frac {\sqrt {a+x^2y^2} -1} {x^2+y^2}} (a>0)$ doesn't exist while a $\ne 1$. How to prove that $\lim_{x\to 0,y\to 0}{\frac {\sqrt {a+x^2y^2} -1} {x^2+y^2}} (a>0)$ doesn't exist while a $\ne 1$?
I already calculated that when a = 1 by multiplying $\sqrt {a+x^2y^2} + 1$ on both denominator and numerator and using the fact $x^2y^2<(x^2+y^2)^2/4$.
Any help will be appreciated.
 [/INST] Evaluating on the path $x=y$:
$$\lim_{x,y\to0}{\frac{\sqrt{a+x^2y^2}-1}{x^2+y^2}}=\lim_{x\to0}{\frac{\sqrt{a+x^4}-1}{2x^2}}=\lim_{x\to0}{\frac{\sqrt a-1}{2x^2}}$$
If $a\ne1$ then $\sqrt a-1\ne0$ and a singularity exists at $x=y=0$, so the limit does not exist.
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