Consider a 2 variable function $f(x,y)$ and the limit
$$\lim_{(x,y)\to (0,0)} f(x,y)$$
If I find two continuous functions $\gamma_1(t)$ and $\gamma_2(t)$ such that $\gamma_1(0)=\gamma_2(0)=(0,0)$ and
$$\lim_{t\to 0} f(\gamma_1(t))=l \in \mathbb{R} \,\,\,\,\,\,\,\,\,\,\, and \,\,\,\,\,\,\,\,\,\, \lim_{t\to 0} f(\gamma_2(t))=+\infty $$
Can I conclude that the original limit $\lim_{(x,y)\to (0,0)} f(x,y)$ does not exist?
I'm not sure about this because of the infinity in one of the two limits above
Yes of course it suffices to find, at least, two different limits finite or infinite, for different paths as for example for
$$\lim_{(x,y)\to (0,0)} \frac{xy}{x^3-y^3} $$
we have
$\frac{xy}{x^3-y^3} \to 0$ for $x=0$
$\frac{xy}{x^3-y^3} \to +\infty$ for $y=-x=t \to 0^+$