Suppose I want to calculate a multivariable limit using substitution. For example, $$ \lim _{(x,y)\to (0,0) } \frac{2^{xy}-1}{xy} $$ or $$ \lim _{(x,y)\to (0,0) } \frac{\sin(xy) }{xy}. $$ Substituting $u=xy$ and then moving to a limit of $u\to 0$ gives a result. As far as I can understand, if the limit of $u\to 0 $ exists, then in particular, the above limits also exist (even though the case of $u\to 0$ can also correspond to e.g. $(x,y)\to(0,2) $).
I am not sure if this is legitimate and if I am right. Is it possible that the substitution $u=xy$ and then $u\to 0 $ only corresponds to a specific path?
Thanks in advance!
Yes in both cases the substitution is legitimate since $u=xy\to 0$ for any path and we obtain
$$\lim _{(x,y)\to (0,0) } \frac{2^{xy}-1}{xy}=\lim _{u\to 0 } \frac{2^{u}-1}{u}$$
and
$$\lim _{(x,y)\to (0,0) } \frac{\sin(xy) }{xy}=\lim _{u\to 0 } \frac{\sin u }{u}$$