However, when x0 is a pole, the order corresponds to the number of terms:

series(1/x^7*exp(x),x=0,3);

        1/x^7 + 1/x^6 + 1/2/x^5 + O(1/x^4)  

@tomleslie Without this "main" condition the problem is trivial: a and x are arbitrary (x<>a) and
m/n = a^2*x^2/(x-a)^2+x^2 

@tomleslie You forgot about the main condition: four integer roots.

restart;
eval(a^2*x^2/(x-a)^2+x^2 = m/n, {a = 21, m = 15616744, n = 73}):
solve(%);

    22, 462, -221 - sqrt(58123), -221 + sqrt(58123)

(isolve is useless here)

@Maple_lover1 Now, for a=1, the two formulae agree for B = r*Pi/k * 2*n/(n-1) but w2 is set to 0 when the argument of sin is not in the interval (-Pi,Pi). Note however that in Maple w2 will be complex when sin(...) < 0 (for n>3).

@mmcdara In the last example, the warning is for g which will be (implicitly) local in the procedure f. X is global in g. Without declaring global X: (in f) , X would be local in g (because you have an assignment X:=X+1). 

@minhthien2016 If you are interested in programming, don't use the document mode.
Add ";" after restart (it should be in a separate execution group).

@tomleslie Thank you. I also had normal results with Usage, but I am puzzled (and the question is) about the ratio t2/t1 and the role of the GUI (with/without output). On a laptop with Maple 2018 I have t2/t1~1.

@Carl Love I was always unhappy with Maple terminology. In math classes, the educators struggle to explain that sin is a function and sin(x) is not; in Maple it's "upside down".

@robertocooper The size and the font of the text can be changed. But Maple is not able to obtain a decent 3D vector graphics (or at least I don't know how; in the versions < 2020 it was even worse). The 2D vector graphics are acceptable. Long time ago, the .eps  for 3D was OK.

@robertocooper 
 

restart:
with(plots):
scheme1 := ["zgradient",["Blue","Cyan","Green","Yellow","Orange","Red"]]:
P1:=plot3d(x*y, colorscheme=scheme1,style=surfacecontour): 
P2:=plot3d(x*y,colorscheme=scheme1,style=point,symbol=asterisk): 
final_plot:=display({P1,P2},size=[2400,2400]);
plottools:-exportplot("abc2400.png",final_plot)

2400.pdf

@nm Unfortuntely the quality of the .eps and .pdf is very low (just zoom into your pdf). It is much better to use a high resolution .png.  It seems that the .eps is obtained by vectorizing a bitmap; this is the reason for its dimension (>3 MB) and quality.
The vector graphics obtained with Maxima are much better and smaller.

@minhthien2016 Hint. Using Pythagorean quadruples (see the wiki article) it's not very difficult to generate (all) such tetrahedrons for which the tangency points with the insphere have rational coordinates. It will suffice to take the center of the insphere in the origin. Try it.

@Carl Love I also don't know why you have presented the quantifiers here. The context seems to be clear: x, y are free variables. Maple is not designed to work with quantifiers. is and coulditbe are practically reserved for the assume facility.
Just try to formalize the definition of the uniform convergence in Maple (F(x,y) --> f(x), for y -->y0), using quantifiers.

@minhthien2016 Replace (in two places) ^+ with ^%T

(but then your version is older than 2018).

@Carl Love I think that it is simpler to consider 
diff( x = 1, x);

which produces the same 1=0.
But, is(x=1) and is(1=0)  are both false. That's OK, because  false implies false  simplifies to true.