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$$ \lim_{x\rightarrow1}(x-1)\lim_{x\rightarrow1}\frac{1}{x-1}$$
I know that the first limit is just $0$
However, I am confused since the second limit diverges.
Should the answer be $0$ (since the first limit is $0$), or should it be undefined? (since the second limit does not exist)
WolframAlpha says it is undefined.
So is it $0$, or undefined?
The expression you wrote is undefined. If you were trying to evaluate: $$ \lim_{x\to 1} \frac{x-1}{x-1} $$ you cannot split into product of limits. $$ \lim_{x\to 1} \frac{x-1}{x-1}\neq \ \lim_{x\to 1} (x-1)\lim_{x\to 1} \frac{1}{x-1} $$
Basic analysis theorems tell you: if two functions $f$ and $g$ admit limits and they are finite, then the limit of the product is the product of the limit.
The same holds for sum, quotient...
The hypothesis are not satisfied, hence you are not allowed to split the limit.
What is your expression? The first factor is $0\in\mathbb R$, but $∞\notin \mathbb R$. The product $0\cdot ∞$ is not well defined, not even inside $\tilde{\mathbb{R}}$, the extended real line, which is a fancy thing, but not a ring where the operation "product" is well defined.
