What is difference in approaches between using standard part in NSA and limit? I don't mean technique of differentiation or integration.
Can somebody explain it on example?
Thanks.
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In NSA, the concept of limit is defined in terms of the standard part. Thus, to calculate the limit of $f(x)$ as $x$ tends to $0$, one would perform two steps: (1) evaluate $f$ at an infinitesimal value $x=\alpha$, and (2) take the standard part. The limit exists if and only if the standard part exists for all infinitesimal $\alpha$, and furthermore they all have the same value.
Mikhail Katz
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Is there case when finding limit is more difficult than finding standard part? – Mike_bb Nov 20 '23 at 18:26
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1Working with limits in non-infinitesimal calculus (using the epsilon-delta definitions) is generally more difficult than working with standard part in infinitesimal calculus. For example, I've had a graduate student (!) tell me that he did not properly understand the notion of uniform continuity until he saw a definition of this notion via infinitesimals (which is simpler than the epsilontic definition). – Mikhail Katz Nov 21 '23 at 10:35
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1@MikhailKatz Though I am not that graduate student, I thought I could let you know that your short answer to Is uniform continuity related to the rate of change of the function? was very useful to me when I has asked the question (as a graduate student). :) – The Amplitwist Nov 27 '23 at 21:39