Limit of cosine function Can I evaluate the following limit
$\lim_{(x, y) \to (0, 1)}\cos (x)$
as below
$\lim_{(x, y) \to (0, 1)}\cos (x)=\lim_{x \to 0}\cos (x)=\cos (0)=1?$
Can I further explain why I can evaluate the limit in that way as follows?
As $(x, y) \to (0, 1)$, $x \to 0$ and $y \to 1$. The cosine function is continuous on the interval $(-\infty, \infty)$. Thus, $\lim_{(x, y) \to (0, 1)} \cos (x)=\lim_{x \to 0} \cos (x)=\cos (0)=1.$

It is always true that:
$$
\lim_{(x_1,\dots,x_n)\rightarrow (x^0_1,\dots,x^0_n)}f(x_i)=\lim_{x_i\rightarrow x^0_i}f(x_i)
$$
without any hypothesis on $f$. It is just not a function of other variables.
