$0 \cdot \infty$ object. Does it make sense?

Question

There maybe a mistake in the question but, let that someone asks you to calculate something like this: $$0\cdot \lim_{x\to0}(log(x))$$ with no further information. The assumptions that one makes is just that log is the natural logarithm, $x\epsilon\mathcal{R}$ and generally maybe some assumptions that a first year calculus course would assume. Nothing too complicated.

The question is the following: How do you somewhat rigorously attack this thing?

My thoughts:

If you see this as a whole is an undefined quantity of the type: $0 \cdot \infty$.

If you see it as parts you have a number $0$ and a limit that diverges. Since the limit does not exist (of course we implicitly assume that $\lim_{x\to0^{+}}$) we could not use the multiplication rule.

Hypothetically if we could use the multiplication rule we run into problems of what function's limit should we represent $0$ with. $x$? $x^2$? $x^{1/10}$?

What do I say then about this object? Does it even make sense to ask something like that?

Answer 1

An algebraic expression needs to have all its terms defined to have a meaning.

As

$$\lim_{x\to0}\log x$$ is undefined, the whole expression $0\cdot\lim_{x\to0}\log x$ is undefined.

And we also have $$(\lim_{x\to0} x)(\lim_{x\to0}\log x)$$

undefined, while

$$\lim_{x\to0}(x\log x)=0.$$

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Also note that $0\cdot\infty$ is not an expression but an expression pattern which describes a limit of the form

$$\lim_{x\to a}(f(x)g(x))$$ where

$$\lim_{x\to a}f(x)=0\text{ and }\lim_{x\to a}g(x)=\infty,$$ as in my third example only.

Answer 2

Generally $\lim_{x\to x_0}f(x)$ is not to be taken as a number. This thing has no meaning by itself. The only meaningful expression is $\lim_{x\to x_0}f(x)=y$, as a whole. Here $y\in\mathbb R \cup\{\pm\infty, \text{DNE}\}$. And $a \lim f(x)$ actually means "We already know that $\lim f(x)=c$ is a number, and we want to look at $ac$," plus some convenient edge cases involving $\pm \infty$. So your expression really has no meaning at all.