I'm doing a bunch of exercises involving the $\operatorname{Tor}$ and $\operatorname{Ext}$ functors and often the functors are non-trivial for $i = 0,1$ and become $0$ for $i \geq 2$. Why is this?
My intuition says that it is because the induced long exact sequence becomes exact after the first $\operatorname{Ext}^1$ or $\operatorname{Tor}_1$ term. Is this true?