Past exam of the mathematics course of the University of Oxford / 2017 / Prelims / Mathematics II / 2 / a

A sequence $(a_n)$ has limit of a sequence $L$ ifFor the reciprocal rule, the required condition is $\boxed{L\ne 0}$. Indeed, eventually $|a_n-L|<|L|/2$, so $|a_n|\ge |L|/2>0$. The reciprocals are then defined on this tail, andA finite number of zero initial terms does not affect this tail limit of a sequence; if a reciprocal sequence is required at every index, those terms must be excluded or assigned arbitrary values separately.