% quessingLimitsPitfall.m

% examine  lim x->0 f(x)

% first try to quess by evaluating f(x) at x closer and closer to x=0
  
  format long
  x = [1e-2 1e-4 1e-6 1e-8 1e-10 1e-12 1e-14]'
  f = (tan(x) - x)./x.^3
  
  disp('it seems that f -> 0 as x ->0')
  
   % What happened. Loss of Significat Digits. Look at the values
   % of the numerator and denominator separately. At x=0.001 we have
   % 0.0000000003/0.000000001 = 1/3.  At x=0.00000001
   % we have 0/(1.0 x 10^(-18)) = 0. 
  
  
  pause                                      % press any key to continue
  
  disp('but graphically this does not seem to be the case, the limit seems to be 1/3')
  x = linspace(-1,1,200);
  f = (tan(x) - x)./x.^3;
  plot(x,f)
  pause
  
  disp(' ')
  disp('let Matlab evaluate the limit with its symbolic toolbox')
  syms x                            % make x a symbolic variable 
  limit((tan(x) - x)./x.^3,x,0)
  
  format