# What are well-known weaknesses of CAS/math software?

I'm starting to become more functional in Mathcad, so I wanted to take the opportunity to look into known drawbacks, mathematically speaking, of CAS/computer math systems. For instance, one of the exercises we did in school involved "breaking" TI Derive by asking for the limit of a discontinuous function (at the point of discontinuity, of course). Another example would be p. 13 of this;

So basically, I'm hoping folks will contribute known problems with these kinds of apps, whether from their experience with a particular package or generalized problems inherent in computer algebra. I myself am most concerned with Mathcad and Mathematica, but it would be awesome for people to talk about CAS/computer math in general and drawbacks of various systems (matlab et al) in particular.

Joe

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I removed the tag computability. –  user2468 Feb 26 '12 at 20:16
I imagine this question is a good candidate for being community wiki! –  Harald Hanche-Olsen Feb 26 '12 at 20:19
IMO the scope of this question is too general. Limitations of numerical packages such as MATLAB are way different than limitations of symbolic packages such as Mathematica. –  user2468 Feb 26 '12 at 20:21
@harald - Why thank you, kind sir! blush –  Joe Stavitsky Feb 26 '12 at 20:27
Here's a relevant collection of issues: math.niu.edu/~rusin/known-math/99/calc_errors It's a bit old and some of the bugs mentioned may have been fixed by now, but nevertheless it never hurts to be cautious when doing mathematics with technology. –  Ted Feb 26 '12 at 21:44
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I only tried this on Mathematica 4, so I don't know if the same problem happens in more modern versions. If you ask Mathematica about $$\int_0^\infty (1-e^{-x})^2/x^2 dx,$$ it tells you that the integral does not converge. But if you split the integral, it gives you

$$\int_0^1 (1-e^{-x})^2/x^2 dx\simeq 0.645751,$$

$$\int_1^\infty (1-e^{-x})^2/x^2 dx\simeq 0.740543$$

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Mathematica 6.0 doesn't complain about divergence, but delivers another strange result: Integrate[(1-Exp[-x])^2 / x^2, {x,0,Infinity}] returns (4 I) Pi + Log[4]. The real part $\ln 4$ agrees with the numerical values, but I wonder where that nonzero imaginary part came from! Got to try this with a newer version at work tomorrow... –  Hans Lundmark Feb 26 '12 at 21:13
Alpha can do it wolframalpha.com/input/… –  user2468 Feb 26 '12 at 21:14
Mathematica 8.0 returns Log[4]. –  Hans Lundmark Feb 27 '12 at 8:50