Evaluating Multivariable Limit $\lim\limits_{(x,y) \to (0,2)} \frac{\sin(xy)}{x}$ Question: Evaluate the limit $$\lim\limits_{(x,y) \to (0,2)} \frac{\sin(xy)}{x}$$
My first thought is that the limit looks a lot like the single variable $\lim\limits_{x \to 0} \frac{\sin(x)}{x} = 1$. Regardless of what $y$ is (as long as it is real) $xy \to 0$. Hence I am wrongly concluding that the entire limit evaluates to $1$. I guess the ratio of the convergence is not the same as in the single variable case, hence it may not be 1. However, I am unsure how to evaluate it properly.

Your idea is correct but we need some correction, indeed since $xy\to 0$ we have that
$$\lim\limits_{(x,y) \to (0,2)} \frac{\sin xy}{x}=\lim\limits_{(x,y) \to (0,2)} \frac{\sin xy}{xy}\cdot \frac{xy}{x}=\lim\limits_{(x,y) \to (0,2)} \frac{\sin xy}{xy}\cdot y=1\cdot2=2$$
