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## Main Question or Discussion Point

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##.

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.

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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