Your command says "cut at +-5 for the y-axis"

Try: plot((x^3 - 4*x^2 - 9*x + 36)/(x^3 - 9*x),  x = -3 .. 3);

It seems to dislike floating point numbers.

L:= convert(x2[1], rational);
numboccur(L, 0);

It is non-negative, but not strictly positive. You may look at

minimize(exp(x)+exp(y)+exp(z)+2*exp(-x-y-z), location);

I am rusty on this - but error 53 refers to run time libraries, so it is *.DLL and not *.xla

PS: you do not really want to post 2 very different questions in just 1 thread.

This is more a question of Numerical Analysis, you may start at https://en.wikipedia.org/wiki/Root-finding_algorithms#Roots_of_polynomials (which also mentiones the method suggested by Corless)

Using vv's suggestion to sum up over smaller intervals I get
0.005295 - 0.0009968*I for n=0 and 0.7493 - 0.4808*I for n=1
for epsilon = 1e-4.

My machines needs about 1.5 h for each case.

Find attached a solution using the NAG routines which gives
it in about 20 seconds for each case, giving the same figures.

MP_232754_NAG.mw

Likewise you write down the recursion function in Cents and finally express it in USD

B:=proc(m::nonnegint) # compute in Cents
round( B(m-1)*(1 + 0.0775) + 400000 );
end proc;
B(0):=400000: # initial

B(30)/100.0; # Cents as USD
                           470415.2400

Writing as iterated integral and handling s first I get
-0.331842471004086e-3 and -.663684942008173e-1

Edit: I have not treated the imaginary part correctly:

-0.331842471004085e-3 -0.331842471004085e-3*I and

-0.0663684942008172 - 0.0663684942008172*I

MP_232562_corrected.mw

MP_232562.mw

Look at plot([0*h[2](x), h[1](x)], x = 0 .. 6, y = -.1 .. .1)

PS: to find possible rounding errors do not use interface(displayprecision = 5), only if you are free of bugs

For Windows you may use its Indexer.

(p+1)*(p-1) = p^2 - 1 is zero modulo 3 by "Little Fermat" (except p=3). Hence one factor divides by 3 and being even (3<=p) it divides by 6.

I once filed Luc Devroye, Non-Uniform Random Variates (1986), having "recipes". May be you can find it as well or find a backup for his site http://cgm.cs.mcgill.ca/~luc/rnbookindex.html through wayback machine

Likewise you may try x = 1+xi, y=x + eta, z=y+zeta (see attached file, it is Maple 2017, I do not have Maple 2021)

MP_232070.mw

I guess you mean "how to prove it", here a suggestion

Task:=Int(diff(u(x, y), x)*diff(u(x, y), x, y) +
  diff(u(x, y), x, x)*diff(u(x, y), y), x);
expand(%);
A,B:=op(1,%), op(2, %);

X:=IntegrationTools:-Parts(A, diff(u(x,y),x));
Y:=IntegrationTools:-Parts(B, diff(u(x,y),y));

Task=X+Y; combine(%);
lhs(%) - rhs(%);
combine(%);

From that it follows obviously.

For example you can save a sheet as html and pick images from there (it is easy to find the according subfolder).