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Proof of Sum Law (Limit Laws)
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Proving the sum law of limits rigorously with the epsilon-delta definition.
REMARKS:
If we have a function h, with domain D, where h(x) = f(x) + g(x), we may evaluate its limit by decomposing this function into the sum of the functions f and g, where f and g both have domain D, and limits L and M at point c respectively.
The sum law tells us that the limit of h(x) (also written as (f+g)(x) to emphasize the fact the function can be split into the sum of two other functions) is simply L + 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.
REMARKS:
If we have a function h, with domain D, where h(x) = f(x) + g(x), we may evaluate its limit by decomposing this function into the sum of the functions f and g, where f and g both have domain D, and limits L and M at point c respectively.
The sum law tells us that the limit of h(x) (also written as (f+g)(x) to emphasize the fact the function can be split into the sum of two other functions) is simply L + 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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