LHospitals rule

L'Hôpital's rule in calculus uses derivatives in order to determine many otherwise hard to compute limits. The rule states: if you are trying to determine the limit of some quotient f(x)/g(x) and both the numerator and denominator approach 0 or infinity, then differentiate numerator and denominator and determine the limit of the quotient of the derivatives; if it exists, it will be the same as the original limit.

For example, a case of "0/0":

         sin(x)          cos(x)      1
    lim  ------  =  lim  ------  =  ---  =  1
    x→0    x        x→0    1         1

and a case of "∞/∞":

          √x            1/(2√x)           √x
    lim  -----  =  lim  -------  =  lim  ----  =  ∞
    x→∞  ln(x)     x→∞    1/x       x→∞    2

Sometimes, even limits which don't appear to be quotients can be handled with the same rule:

                               (x + √(x2 - x) (x - √(x2 - x)           x2 - (x2 - x) 
    lim  x - √(x2 - x)  =  lim -----------------------------  =  lim  --------------
    x→∞                    x→∞       (x + √(x2 + x)              x→∞   x + √(x2 + x  

                       x                             1                  1
          =  lim  --------------  =  lim  ------------------------  =  ----  =  1/2
             x→∞  (x + √(x2 + x)     x→∞  1 + (2x + 1)/(2√(x2 + x))     1+1