Proof of Reciprocal Law (Limit Laws)
Автор: Hayashi Manabu
Загружено: 2021-04-24
Просмотров: 3058
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.
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