What is the limit of zero times x, as x approaches infinity?

Note that for any $x$ we have $x\cdot 0=0$ and therefore

$$\lim_{x\to\infty} (x\cdot 0) =\lim_{x\to\infty} 0=0$$

As others have said, $\lim_{x\to \infty} 0 \times x = \lim_{x\to\infty} 0 = 0$. I'm going to expand a bit more on "$0 \times \infty$ is undefined".

We can't do operations with $\infty$ directly, as you know. But we can do operations with "functions with limit $\infty$", and if they behave well enough then that might give us reasonable definitions of things like "$0 \times \infty$".

However, if we replace $\infty$ by "functions with limit $\infty$" then we ought to do the same with $0$, i.e. replace $0$ by "functions with limit $0$". This is a reasonable thing to do, because it works for real numbers:

For any functions $f(x)$ and $g(x)$ such that $\lim_{x \to \infty} f(x) = a$ and $\lim_{x \to \infty} g(x) = b$, where $a$ and $b$ are real numbers (i.e. finite), it's the case that $\lim_{x \to \infty} (f(x)g(x)) = ab$.

In fact, it's better than that; if $a > 0$ is real then it's reasonable to say "$a \times \infty = \infty$":

For any functions $f(x)$ and $g(x)$ such that $\lim_{x \to \infty} f(x) = a$, where $a$ is a real number and $a > 0$, and such that $\lim_{x \to \infty} g(x) = \infty$, then $\lim_{x \to \infty}(f(x)g(x)) = \infty$ also.

However, if $f(x)$ and $g(x)$ are such that $\lim_{x \to \infty} f(x) = 0$ and $\lim _{x \to \infty} g(x) = \infty$, then we don't know anything about $\lim_{x \to \infty} f(x)g(x)$. If $g(x) = x$, then taking $f(x) = 0$, $f(x) = a/x$ (where $a > 0$), and $f(x) = 1/\sqrt x$, gives $\lim_{x \to \infty} (f(x)g(x))$ to be, respectively, $0$, $a$ and $\infty$. This is why we say "$0 \times \infty$" is undefined.

For every $x\in \mathbf R$ it holds $0 \cdot x = 0$. Using this definition, we let $f(x):= 0 \cdot x = 0$ for every $x\in \mathbf R$. Hence, we have $$ \lim_{x\to +\infty } x\cdot 0 = \lim_{x\to +\infty } f(x) = \lim_{x\to +\infty }0 =0$$ from the definition of the limit! There is no need to think about something like $0 \cdot (+\infty)$.