L'Hopital Rule

GWM For evaluation of limits. Eg see Simple method for computing mathematical limits. In many operations we find rational functions.

           f(x)     f'(0)
   lim    ------ = ------
   x->0     g(x)    g'(0)

evaluating rational functions at certain values can give problems, such as what is the result of dividing 0 by 0 in this:

   lim    -------  (actually it is = 1!)
   x->0      x

L'Hopital's rule is "The limit of a rational function can be found by replacing the top and bottom function by their derivatives." [L1 ]

For the sin(x)/x case, d/dx(sin(x))=cos(x); d/dx(x)=1. So the limit is cos(0)/1 = 1. Other examples:

   lim    ------ = k (= k.cos(0))       
   x->0     x                         

         (1-cos(x)^2)    1  {   sin(0)   cos(0) }
  lim    ------------ = --- {= ------ = ------  }
  x->0       x^2         2  {  2.[x=0]    2     }

   lim    ------ = 1/k (= cos(0)/k)
   x->0     k.x

IF the ratio of derivatives is also 0/0, use the second derivative and so on as in the second case above. [Simplified proof of L'Hôpital: if the slope of the top part of fraction is N times the slope of the bottom part then the ratio of the values near the zero value is N (go to dx, top part has value N.dx, bottom value is dx).] The L'Hôpital rule can also be applied at infinity, where the numerical method should have a problem:

   lim    ------ = Inf (= exp(Inf)/1 since d/dx(exp(x))=exp(x) & d/dx(x)=1)
   x->Inf     x

Application of L'Hopital rule may save considerable effort in coding a means of evaluating the limit.