Limit exists, it is correct. In more detail, if you factor the numerator,

x^2-2 = (x - sqrt(2))*(x+sqrt(2))

and then cancel the (x-sqrt(2) term, you get x+sqrt(2). The function is equal to that in every point except x=sqrt(2). That's why the limit exists. It is not "undefined". And it can be seen from the plot.When x is approaching sqrt(2), y is approaching 2*sqrt(2) which is the limit.

The plot in Maple is constructed from points - in this example, 49 points, connected with line segments. The x-coordinates of points missed value x=sqrt(2), and there are no any problems in other points. So Maple connected those 49 points and got a straight line.

The correct plot should look as in the example I posted above.

Alec

I already said about that - there should be 2 things for continuity - the function should be defined, the limit should exist, and the value of the function should be equal to the limit.

In this case, the limit exists, but the function is not defined, so it is a discontinuity.

Maple has a function discont for finding discontinuities. It works well in this example,

discont((x^2-2)/(x-sqrt(2)),x);
                                  1/2
                                {2   }

If you do a plot, this point should be represented on it as an empty circle. Something like that,

with(plots): 
a := plot((x^2-2)/(x-sqrt(2))): 
b := pointplot([sqrt(2), 2*sqrt(2)+0.05], 
    symbolsize = 15, symbol = circle, color = blue): 
c := pointplot([sqrt(2), 2*sqrt(2)+0.05], 
    symbolsize = 10, symbol = solidcircle, color = white): 
display(a, b, c);

Alec

Yes.

Publishing is a complicated process, especially in math, so such things happen from time to time. Authors usually put errata (a collection of known typos) on the book web pages if such pages exist.

Alec

It is not defined at x=sqrt(2). All rational functions are continuous in every point they are defined, including x=2 in this example.

To be continuous at some point x=a, a function f should satisfy 2 conditions:

a) be defined at that point;

b) the limit of f at that point should exist and should be equal to f(a).

In your example, the function is not continuous at x=sqrt(2) because the condition a) is not satisfied - it is not defined there.

Such discontinuities, with existing limit, are called 'removable' - the function becomes continuous (at that point) after extending its definition to being equal to the limit at that point.

Alec

You might try unbind instead of unwith. Also, restart command restarts Maple.

Alec

The solutions in this case (order 2) have form y(t)=(a+bt)/(c+dt) or y(t)=a+bt+ct^2. That gives a conjecture that for Runge-Kutta of order n the solutions have form y(t)=f(t)/g(t) with f(t) and g(t) being polynomials with sum of degrees less or equal than n. There may be some additional restrictions on the degree of the denominator (say, being less or equal than n/2 or floor of (n+1)/2, or something like that), because 1/f(x) with f(x) of degree 2 didn't appear in the list of solutions for Runge-Kutta of order 2. 

That seems to be working for order 2n with solutions y(t)=f(t)/g(t) with f(t) and g(t) being polynomials of degree n. The differential equation in this case can be obtained by differentiating g(t)y(t)=f(t). But I didn't check all the details and I didn't check other cases.

Alec

And what after that - implement wedge product for Arrays myself? No doubt that I can do that in a couple of minutes (as well as many other people). I am not talking about using it myself - I don't use Maple. It is just the general development process seems to be wrong.

In particular, the first step - specifications and requirements seem to be missing. As well as review and testing.

Alec

That seems to lead to complicated algebraic equations for higher orders, as in that post?

Alec

Being surprised by an output of solve with inequalities, I looked at the ?solve,ineqs page. As usual, a typo - solvefrom .

Alec

GAP is included in SAGE, as well as many other things.

Alec

When I used to use it, Java usually keeped deprecated features for 2-3 releases (actually, longer than that) after they were deprecated. How large is the difference between the current release and the release used in Maple?

I should say that I was pleasantly surprised that Standard Maple worked in Vista with Aero without changing to the compatibility mode, as many other Java applications do. That was done well.

Alec

When I used to use it, Java usually keeped deprecated features for 2-3 releases (actually, longer than that) after they were deprecated. How large is the difference between the current release and the release used in Maple?

I should say that I was pleasantly surprised that Standard Maple worked in Vista with Aero without changing to the compatibility mode, as many other Java applications do. That was done well.

Alec

f:=add(add(x||i*x||j,i=1..j),j=1..20):
time(expand(f^2));

                                0.031

t:=time():
g:=add(add(x||i*x||j,i=1..j),j=1..100):
time()-t;

                                  0.

time(expand(g^2));

                                82.477

with(DifferentialGeometry):
DGsetup(['x||i'$i=1..20],M):
F:=evalDG(add(add(dx||i &w dx||j,i=1..j),j=1..20)):
time(evalDG(F &w F));

                                28.002

DGsetup(['x||i'$i=1..100],N):
t:=time():
G:=evalDG(add(add(dx||i &w dx||j,i=1..j),j=1..100)):
time()-t;

                               485.600

time(evalDG(G &w G));
Warning, computation interrupted

Alec

Talking about Grassmann algebra, I meant differential forms. It's one of names for them. Another name is exterior algebra. Such things as 1+dx+2*dy -3*dx ∧ dy with addition of them, multiplication by scalars and ∧ operation, defined for all of them.

DifferentialGeometry package doesn't implement that. I didn't look at things called "tensors" there. Mainly because the documentation is also not that great. Are such things as 1+ x+2*y - 3*x⊗y also not defined and ⊗ is defined not for any 2 "tensors", but only for some of them?

My opinion about such things is the same as Yoda's - "Do, or do not. There is no try." Translating to English - do things the right way, or don't do them at all.

That reminded me about another thing. Teaching Java 8-10 years ago, I used to tell students that it is so easy language, everybody can program in it. One doesn't have to be a programmer to do that... Well... Some software applications which may be related or not related to this site clearly proved me wrong. Not everybody.

Alec

LyX also. It is not a pure LaTeX, but it is derived from it and is close to it. There are plugins for both Maple and Sage in TeXmacs and they also can be accessed from LyX (but not that easily.)

I would like both of them more if they used pure LaTeX. Lyx is closer to that. Originally, it started as WySiWym front-end for LaTeX, but then it started to go further and further from that. Still, it is much closer to LaTeX than TeXmacs.

Also, TexMacs's second part of the name which came from emacs, shows how it uses keyboard - a lot of commands need to push 3 keys simultaneously, as in emacs. There are people who like that, but I am not one of them.

Alec