Infinitesimals Rajan Jha

Understanding infinitesimals rajan jha requires examining multiple perspectives and considerations. What is the meaning of infinitesimal? - Mathematics Stack Exchange. Infinitesimals provide an alternative approach that is more accessible to the students and does not require excursions into logical complications necessitated by the epsilon, delta approach. How do you understand Infinitesimals? On the other hand, the hyperreal number system $ {}^ {\ast}\mathbb R$ contains both infinite numbers and infinitesimals, and there you can rigorously speak about a number with an infinite tail of $3$'s falling infinitesimally short of $\frac13$, or of a number with an infinite tail of $9$'s falling infinitesimally short of $1$, precisely as you ...

Are infinitesimals still being used in calculus?. Moreover, infinitesimals are banished from a standard rigorous development of calculus, because it's difficult to make them precise. But "infinitesimal intuition" is simple, clear, makes calculus seem obvious, and is at the heart of understanding calculus, in my opinion. What's an example of an infinitesimal? From another angle, if you want to use infinitesimals to teach calculus, what kind of example of an infinitesimal can you give to the students?

In this context, what I am asking for are specific techniques for explaining infinitesimal... Are infinitesimals equal to zero? Meni, your comments on infinitesimals are well taken, but your comments about limits reveal a common misconception. Limits can be defined either via epsilon-delta or via infinitesimals.

In the latter case one exploits the standard part function, as explained in numerous posts under tags infinitesimals and nonstandard-analysis. Why infinities but not infinitesimals? Infinitesimals are entirely possible to avoid. Maybe this difference in necessity that makes people more comfortable dealing with infinities than infinitesimals? Newest 'infinitesimals' Questions - Mathematics Stack Exchange.

For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also [nonstandard-analysis]). Furthermore, integral Calculus, Infinitesimal - Mathematics Stack Exchange. A theorem like "the (defined-in-terms-of-infinitesimals) integral of a continuous real function over a closed interval has a unique finite real value" is proved in the integration chapter of Keisler's Elementary Calculus.

And at a more advanced level, Terry Tao uses infinitesimals to prove theorems often (see his NSA blog posts). What's so different about limits compared to infinitesimals?. Ask Question Asked 9 years, 4 months ago Modified 8 years, 6 months ago Do limits leave residual infinitesimals, or do they resolve exactly?.

Thus, "residual infinitesimals" are indeed involved in evaluating a limit, but they disappear once we apply the standard part. For an elementary axiomatic approach to infinitesimal analysis (that does not involve either the axiom of choice or ultrafilters), see this introduction.