So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where ##n>1##) is equal to 0, leaving only the relation between the 1st order term ##x## and ##y##.(adsbygoogle = window.adsbygoogle || []).push({});

For example in a Poisson process the chance of arrival in a time interval ##δ## where ##δ→0## , is ##λδ## (where ##λ## is the arrival frequency). The chances of no arrivals during said interval is ##1-λδ## and the chance of 2 arrivals or more is 0, because the chance of getting n arrivals in the interval ##δ## is ##(λδ)^n=λ^nδ^n## and ##δ^n=0## for ##n>1## when ##δ→0##.

Now in the basic intuitive sense I can understand why this is the case, if a variable ##x## approaches 0 then the variable ##x^2## (or ##x^n## where n>1) becomes negligibly small, and it becomes more and more negligible as ##x## becomes smaller and smaller. The thing is we are already dealing with infinitesimals in cases like the Poisson process, so why do we decide that ##x## is not negligible and ##x^2## is when both are arbitrarily small?

I guess I'm asking for a mathematical basis for this claim, I'm sure there is one since it is so confidently used in many fields in math and physics.

Thanks in advance to all the helpers.

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# I Differentials of order 2 or bigger that are equal to 0

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