How can I solve $\lim_{(x,y)\to (1,2)}\frac{(x-1)(y-2)}{(x-1)^2+\sin^2(y-2)}$

Question

I'm trying to solve the limit below,

$$\lim_{(x,y)\to (1,2)}\frac{(x-1)(y-2)}{(x-1)^2+\sin^2(y-2)}$$

Just try to solve this limit and I don't know how to approach to this kind of questions when the denominator and the numerator are tending both to 0 when i subtitute $x=1$ and $y=2$. I tried to use trigonemetric identity for making the expression more simple and than to solve it but it doesn't help. As i tryed other question like this i succeed by making the expression more simple and than found someway to calculate it but i am not sure its the way to approach this kind of questions.

Answer 1

We have

$$\begin{align}\lim_{(u,v)\to (0,0)}\frac{uv}{u^2+\sin^2 v}&=\lim_{(u,v)\to (0,0)}\frac{\frac uv}{\left(\frac uv \right)^2+1}\\ &=\lim_{(u,v)\to (0,0)}\frac{uv}{u^2+v^2}.\end{align}$$

Then $u=r\sin x ,~ v=r\cos x$

$$\begin{align}\lim_{(u,v)\to (0,0)}\frac{uv}{u^2+v^2}&=\lim_{r\to 0}\frac{\frac 12r^2 \sin (2x)}{r^2}\\ &=\frac 12 \sin (2x).\end{align}$$

This shows that, limit doesn't exist.