% script e10maple.m
%
% if you know about limits, you should be able to apply the limit test
% (posted to newsgroup; you might want to use L'Hopital's Rule on E10.8).
%
% however, if you don't know about limits or have trouble computing a
% limit, then often matlab can help.  this script is to show you how
% to use the limit test and how to use some features of the matlab's
% symbolic math toolbox.
%
% note: the symbolic math toolbox might not be part of student versions
%       of matlab; if so, go to a public cit lab.
% 
% matlab includes something called $maple$ (see $help symbolic$) that can do 
% *symbolic manipulation* (computation with mathematical expressions),
% as distinct from *numerical computation*" (computation with numbers).
% 
% this is bonus material for CS100M this semester, 
% but you might find it very helpful for CS100M and other courses!
% 
% =============================================================================
% 
%                                                             f(n)
% recall how to interpret the result of the limit test  lim   ----
%                                                      n->oo  g(n)
% for comparing O(f(n)) to O(g(n)):
%
% limit      test diagnosis
% -------    -----------------
% 0          O(f(n)) < O(g(n))
% ]0,inf[    O(f(n)) = O(g(n))
% inf        O(f(n)) > O(g(n))
% does not
% exist      test is "inconclusive" (depends on how/why limit does not exist)
% 
% let's apply the test to E10.4 - E10.8
%
% note: matlab tries to simplify fractions, 
%       so f(n)/g(n) won't always look as you might expect.
%

help e10maple % print out comments above

disp('press any key to run the tests...'); pause

clear

syms n % declare $n$ to be a "symbolic variable"

sep(1:70) = '*'; sep = sprintf('%s\n\n', sep);

label = sprintf('\n\nlim n --> oo of');


disp([sep 'E10.4. O(100 n^2 + 5n + 1) versus O(n^2 + 5n + 100)'])

    ratio = [100 * n^2 + 5*n + 1] / [n^2 + 5*n + 100];
    disp(label), pretty(ratio), disp('is'), limit(ratio, n, inf)

disp([sep 'E10.5. O(10 n^2) versus O(2 n^10)'])

    ratio = [10 * n^2] / [2 * n^10];
    disp(label), pretty(ratio), disp('is'), limit(ratio, n, inf)

disp([sep 'E10.6. O(10) versus O(1)'])

    ratio = [10] / [1]; 
    disp(label), '10/1', disp('is'), limit(ratio, n, inf)

disp([sep 'E10.7. O(1 + 1/n) versus O(2)'])
    ratio = [1 + 1/n] / [2];
    disp(label), '(1+1/n) / 2', disp('is'), limit(ratio, n, inf)

disp([sep 'E10.8. O(n+log(n)) versus O(n)'])

    ratio = [n + log(n)] / [n];
    disp(label), pretty(ratio), disp('is'), limit(ratio, n, inf)

disp([sep 'example from newsgroup: 2 - (-1)^n versus 2 + (-1)^n'])
    ratio = [2 - (-1)^n] / [2 + (-1)^n];
    disp(label), pretty(ratio), disp('is'), limit(ratio, n, inf)
    disp('note: the 1/3 .. 3 means maple/matlab found 2 possible answers for')
    disp('the limit --namely 1/3 and 3-- which means the limit does not exist')