Limits of functions in SMath.

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Best regards.
Alvaro.

Very good to have a native limit function in SMath Studio. I was curious how Maxima could handle the examples.

Anyways thanks for the idea of indicating left and right side limit using exponentiation with a special character. That prevents unintended pre-processing as it happened with the original trick of adding +0 or -0. An alternative would be to use a "unit zero", i.e. +'0 and -'0.
I think that neither the unit zero nor the special character exponent method are obvious for the user.

Perhaps best would be to have three limit functions: lim(), limr() and liml() with the side indicator being part of the glyphs.

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Wrote

Perhaps best would be to have three limit functions: lim(), limr() and liml() with the side indicator being part of the glyphs.

Yes, I think that too. A note about the differences where Maxima assumes a positive for the result. Notice that in Maxima ∞ means +∞, but the idea in the SMath lim function is that ∞ means "unsigned infinitum", which is the true equivalent for 1/0, when 1/0+ = +∞ and 1/0- = -∞.



Let me explain.

What's the sense for distinguish the cases 0, 0+, 0- if you don't do the same thing with ∞? This is: if 1/x -> ∞ for x -> 0, why 1/x -> ∞ for x -> 0+, that's wrong: ∞ = 1/0 but ∞ = 1/0+. There are 3 infinity's in the real line: plus, minus and unsigned, which means infinity for both sides at the same time, like what happens with 1/x at x = 0. And because Maxima have two infinity's, not three, is why can't calculate the last lim(x*sin(x),x,∞)=und which is not +∞, nor -∞ but "∞", this is, unsigned ∞ which grows to infinity for "both sides": to +∞ and -∞ at the same time, like when 1/x decrease to zero for 0+ and 0- at the same time.



So Maxima have troubles for don't recognize the unsigned infinity: make continuous assumptions about the sign of the constants or directly can't give an answer for very simple examples. The infinitum in the lim function is an unsigned infinitum. Notice that in the complex plane the situation is more chaotic, you can grow to infinity in many distinct ways +∞ -∞*i, ∞ - ∞*i (unsigned in real axis, negative in the imaginary) ... etc.

Best regards.
Alvaro.

Actually it is not Maxima making the assumptions on the sign of variables or expressions. In an interactiv Maxima session, in such cases the user is asked to provide additional information. Yet the plugin catches these questions and answers them with "pos" by default, because this is what you want in most cases, just to mention units. In order to make this transparent, an according message is bundled to the result. If you want to get rid of it, you can be specific using "assume().

I didn't know that 1/(x sin(x)) is said to converge to 0 for x -> inf. That must be sort of higher order understanding of limits, where a nonzero chance to find anything outside the eps-environment of the limit is accepted, if the limit of that chance is zero.

I don't think that using the infinity symbol for the set of pos and neg infinity is practical in SMath studio or on paper unless you insist on unary + operators being significant and a being something different from +a. I guess that would surprise most users.

Rather I'd recommend to use +-inf, given that +- is available as a set-generator already.

Wrote

Actually it is not Maxima making the assumptions on the sign of variables or expressions. In an interactiv Maxima session, in such cases the user is asked to provide additional information. Yet the plugin catches these questions and answers them with "pos" by default, because this is what you want in most cases, just to mention units. In order to make this transparent, an according message is bundled to the result. If you want to get rid of it, you can be specific using "assume().

Yes, I wrote that for comparing with the SMath default's assumptions for the "infinity's algebra" (for calling that with something)

Wrote

I didn't know that 1/(x sin(x)) is said to converge to 0 for x -> inf. That must be sort of higher order understanding of limits, where a nonzero chance to find anything outside the eps-environment of the limit is accepted, if the limit of that chance is zero.

Yes, you're absolutely right. Sorry, I don't know about what I could be thinking at that moment.

Wrote

I don't think that using the infinity symbol for the set of pos and neg infinity is practical in SMath studio or on paper unless you insist on unary + operators being significant and a being something different from +a. I guess that would surprise most users.

Rather I'd recommend to use +-inf, given that +- is available as a set-generator already.

You're right again about it is practical in a CAS or not. But it is not related about the unary operators + or -, it is related with the topological concept of path in a domain. For reals you have three paths for go to zero: by left, by right or by both sides at the same time. All CAS can handle that very well, but have troubles handling the inversion of that: 1/0+ = +∞ , 1/0- = -∞ but 1/0 = ∞ unsigned. For example, I don't see what the sense of not recognize this limit:



Also I don't think that assign that result to infinity could surprise anyone. In the classic literature also you have unsigned infinity everywhere.

Rey Pastor, Análisis Matemático vol I, page 274 first introduce unsigned infinity limit for successions and +∞ -∞ as particular cases



Piskunov, Differential and Integral Calculus, page 53 take an entire page for demonstrate that a function converges for unsigned infinity and put that as a theorem.


That could be too much for real calculus, but I guess that it is the preamble for work with the complications of the topology of the complex plane in a further study of limits. So, I agree with you that it could be impractical to introduce it in SMath, but also what I see are unnecessary complications without it.

Best regards.
Alvaro.