Is $ \lim_{n \to \infty} (n \cdot 0) = 0 $ or undefined?

Question

I am trying to learn how to work with infinities correctly while using limits. I don't think I have understood all the rules correctly yet because I am getting a contradiction:

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

But I also got:

$$ \lim_{n \to \infty} (n \cdot 0) = (\lim_{n \to \infty} n) \cdot (\lim_{n \to \infty} 0) = \infty \cdot 0 = \text{undefined}.$$

Both can't be right. Which one is correct and why? What rule did I violate in the incorrect one?

Since the expression is $0$ for every positive integer $n$ , the limit is $0$. — Peter, Dec 30 '22 at 13:20
@Peter So the second equation is incorrect. Which step in that is incorrect? — Lone Learner, Dec 30 '22 at 13:22
That the limit of the product is equal to the product of the limits only holds if both limits exist. — Peter, Dec 30 '22 at 13:22

score 7 · Accepted Answer · answered Dec 30 '22 at 13:22

The equality$$\lim_{n\to\infty}(a_nb_n)=\left(\lim_{n\to\infty}a_n\right)\left(\lim_{n\to\infty}b_n\right)\label{a}\tag1$$holds assuming that all limits involved exist and that the product from the RHS of \eqref{a} is defined (in $\Bbb R\cup\{\pm\infty\}$). That's not the case in your second approach. The first one is correct.

Is $ \lim_{n \to \infty} (n \cdot 0) = 0 $ or undefined?

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