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Suppose we have a limit: $$ \lim_{x\rightarrow 1}\frac{x-1}{x^2-1}=\lim_{x\rightarrow 1}\frac{x-1}{(x-1)(x+1)}=\lim_{x\rightarrow 1}\frac{1}{x+1}=0.5 $$ Here we can cut $(x-1)$ because $x\rightarrow 1$ so $x\neq 1$ and $(x-1)\neq 0$.

When in general can we cut same terms in the numerator and the denominator (in limits, in equations etc) and when we can't?

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    It holds for all $a, b\in \mathbb R, a, b \neq 0$ that $\frac{a}{ab} = \frac{1}{b}$ – 2017-02-13

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Your reason for cutting the term in the limit seems more like a motivation for cutting the term. Even if it does not turn the limit into something that is not an indeterminate form you can still do it. This is a fundamental axiom of algebraic manipulations that you can always cancel denominators and numerators if they are the same. Even if they are in limits.

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    L'Hopitals Rule will give the same result ... that hopefully will go some way to convince you that it is perfectly legitimate to cancel the $x-1$. – 2017-02-13
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    The reply sounds strange to me, because 0/0 is indetermined... – 2017-02-14
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    You can divide (that's a better word than "cut") top and bottom by exactly the same nonzero (and that's important: nonzero!) term. And $x-1$ is nonzero for every $x$ not equal to $1$, $\lim_{x\rightarrow 1}\frac{x-1}{(x-1)(x+1)}=\lim_{x\rightarrow 1}\frac{1}{x+1}$ is legitimate. – 2017-02-26