Proof of Reciprocal Law (Limit Laws)

Proving the reciprocal law of limits rigorously with the epsilon-delta definition.

REMARKS:
If we have a function h, with domain D, where h(x) = 1 / g(x), we may evaluate its limit by considering the function g, with domain D (note that g(x) would never be 0 in D since D is the domain of h).

The reciprocal law tells us that the limit of h(x) (also written as (1/g)(x) to emphasize the fact the output of the function is the reciprocal of the output of g) is simply 1 / M.

Of course all of this goes without saying that we've assumed point c to be a cluster/accumulation point of the domain D. It makes no sense to talk about the limit if this condition wasn't satisfied.

This law can help us establish a whole range of results without having to go through the hassle of establishing an epsilon-delta definition all the time.