How to solve the limit $\lim_{(x,y)\to(2,3)}\frac{\sin(x)+2}{(x^2-y^2+5)^2}$ How can I solve the following limit or prove that the limit does not exist in $\Bbb R$:
$$
\lim_{(x,y)\to(2,3)}\frac{\sin(x)+2}{(x^2-y^2+5)^2}?
$$
I tried to prove the limit does not exist with $y=3x/2$, $y=x+1$, but it looks like a dead end.

Note that since $-1 \le \sin x \le 1$, we have:
$$\frac{1}{\left(x^2-y^2+5\right)^2} \le \frac{\sin x +2}{\left(x^2-y^2+5\right)^2} \le \frac{3}{\left(x^2-y^2+5\right)^2}$$
And the denominator clearly tends to $0$ (but stays positive) when $(x,y) \to (2,3)$, so...
